Layered Architecture for Quantum Computing

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page PHYSICAL REVIEW X 2, 031007 (2012)

Layered Architecture for Quantum Computing[edit]

N. Cody Jones,1,* Rodney Van Meter,2 Austin G. Fowler,3 Peter L. McMahon,1 Jungsang Kim,4 Thaddeus D. Ladd,1,5 and Yoshihisa Yamamoto1,5

1 Edward L. Ginzton Laboratory, Stanford University, Stanford, California 94305-4088, USA
2 Faculty of Environment and Information Studies, Keio University, 5322 Endo, Fujisawa 252-0882, Japan
3 Centre for Quantum Computation and Communication Technology, University of Melbourne, Victoria 3010, Australia
4 Fitzpatrick Institute for Photonics, Duke University, Durham 27708, North Carolina, USA
5 National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan

(Received 1 September 2011; revised manuscript received 5 April 2012; published 31 July 2012)

We develop a layered quantum-computer architecture, which is a systematic framework for tackling the individual challenges of developing a quantum computer while constructing a cohesive device design. We discuss many of the prominent techniques for implementing circuit-model quantum computing and introduce several new methods, with an emphasis on employing surface-code quantum error correction. In doing so, we propose a new quantum-computer architecture based on optical control of quantum dots. The time scales of physical-hardware operations and logical, error-corrected quantum gates differ by several orders of magnitude. By dividing functionality into layers, we can design and analyze subsystems independently, demonstrating the value of our layered architectural approach. Using this concrete hardware platform, we provide resource analysis for executing fault-tolerant quantum algorithms for integer factoring and quantum simulation, finding that the quantum-dot architecture we study could solve such problems on the time scale of days.

DOI: 10.1103/PhysRevX.2.031007

Subject Areas: Quantum Physics, Quantum Information

* ncodyjo???

Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.


Quantum computing as an engineering discipline is still in its infancy. Although the physics is well understood, developing devices that compute with quantum mechanics is technologically daunting. While experiments to date manipulate only a handful of quantum bits [1], we consider what effort is required to build a large-scale quantum computer. This objective demands more than a cursory estimate of the number of qubits and gates required for a given algorithm. One must consider the faulty quantum hardware, with errors caused by both the environment and deliberate control operations; the classical processing required when error correction is invoked; the special treatment required for constructing arbitrary gate sequences from a limited fault-tolerant set; and so on. This paper provides a framework for addressing the complete challenge of designing a quantum computer. Many researchers have presented and examined components of large-scale quantum computing. We study here how these components may be combined in an efficient design, and we introduce new methods that improve the quantum computer we propose. This engineering pursuit is quantum-computer architecture, which we develop here in layers. An architecture decomposes complex system behaviors into a manageable set of operations. A layered architecture does this through layers of abstraction where each layer embodies a critical set of related functions. For our purposes, each ascending layer brings the system closer to an ideal quantum-computing environment. The paper is organized as follows. The remainder of Sec. I provides a global view of a layered quantumcomputer architecture, indicating how the topics we examine are connected to each other. Section II enumerates the essential components of a quantum computer by examining a new hardware platform based on the optical control of quantum dots. Section III discusses control techniques for suppressing hardware errors prior to using active error correction. Section IV demonstrates how to implement and account for the resources of quantum error correction, with particular emphasis on the surface code [2]. Section V analyzes the necessary techniques for constructing universal quantum gates from the limited set of operations provided by error correction. Section VI calculates the computer resources necessary to implement two prominent quantum algorithms: integer factoring and quantum simulation. Section VII discusses timing issues that affect how the layers in the architecture interact with each other. Finally, Section VIII discusses how our findings are applicable to future work in quantum computing.

A. Prior work on quantum-computer architecture[edit]

Many different quantum-computing technologies are under experimental investigation [1], but for each a scalable system architecture remains an open research page 1 problem. Since DiVincenzo introduced his fundamental criteria for a viable quantum-computing technology [3], and Steane emphasized the difficulty of designing systems capable of running quantum error correction (QEC) adequately [4,5], several groups of researchers have outlined various additional taxonomies addressing the architectural needs of large-scale systems [6,7]. As an example, small-scale interconnects have been proposed for many technologies, but the problems of organizing subsystems using these techniques into a complete architecture for a large-scale system have been addressed by only a few researchers. In particular, the issue of heterogeneity in system architecture has received relatively little attention. The most important subroutine in fault-tolerant quantum computers considered thus far is the preparation of ancilla states for fault-tolerant circuits, because these circuits often require very many ancillas. Taylor et al. proposed a design with alternating ‘‘ancilla blocks’’ and ‘‘data blocks’’ in the device layout [8]. Steane introduced the idea of ‘‘factories’’ for creating ancillas [9], which we examine for the case of the surface code in this work. Isailovic et al. [10] studied this problem for ion-trap architectures and found that, for typical quantum circuits, approximately 90% of the quantum computer must be devoted to such factories in order to calculate ‘‘at the speed of data,’’ or where ancilla production is not the ratelimiting process. The findings we present here are in close agreement with this estimate. Metodi et al. also considered production of ancillas in ion-trap designs, focusing instead on a 3-qubit ancilla state used for the Toffoli gate [11], which is an alternative pathway to a universal fault-tolerant set of gates. Some researchers have studied the difficulty of moving data in a quantum processor. Kielpinski et al. proposed a scalable ion-trap technology utilizing separate memory and computing areas [12]. Because quantum error correction requires rapid cycling across all physical qubits in the system, this approach is best used as a unit cell replicated across a larger system. Other researchers have proposed homogeneous systems built around this basic concept. One common structure is a recursive H tree, which works well with a small number of layers of a Calderbank-ShorSteane code, targeted explicitly at ion-trap systems [13,14]. Oskin et al. [15], building on the Kane solid-state NMR technology [16], proposed a loose lattice of sites, explicitly considering the issues of classical control and movement of quantum data in scalable systems, but without a specific plan for QEC. In the case of quantum computing with superconducting circuits, the quantum von Neumann architecture specifically considers dedicated hardware for quantum memories, zeroing registers, and a quantum bus [17]. Long-range coupling and communication is a signifi- cant challenge for quantum computers. Cirac et al. proposed the use of photonic qubits to distribute entanglement between distant atoms [18], and other researchers have investigated the prospects for optically mediated nonlocal gates [19–23]. Such photonic channels could be utilized to realize a modular, scalable distributed quantum computer [24]. Conversely, Metodi et al. consider how to use local gates and quantum teleportation to move logical qubits throughout their ion-trap Quantum Logic Array architecture [11]. Fowler et al. [25] investigated a Josephson-junction flux-qubit architecture considering the extreme difficulties of routing both the quantum couplers and large numbers of classical control lines, producing a structure with support for CalderbankShor-Steane codes and logical qubits organized in a line. Whitney et al. [26,27] have investigated automated layout and optimization of circuit designs specifically for ion-trap architectures, and Isailovic et al. [10,28] have studied interconnection and data-throughput issues in similar ion-trap systems, with an emphasis on preparing ancillas for teleportation gates [29]. Other work has studied quantum-computer architectures with only nearest-neighbor coupling between qubits in an array [30–34], which is appealing from a hardware design perspective. With the recent advances in the operation of the topological codes and their desirable characteristics such as having a high practical threshold and requiring only nearest-neighbor interactions, research effort has shifted toward architectures capable of building and maintaining large two- and three-dimensional cluster states [35–38]. These systems rely on topological errorcorrection models [39], whose higher tolerance to error often comes at the cost of a larger physical system, relative to, for example, implementations based on the Steane code [40]. The surface code [2], which we examine in this work for its impact on architecture, belongs to the topological family of codes. Recent attention has been directed at distributed models of quantum computing. Devitt et al. studied how to distribute a photonic cluster-state quantumcomputing network over different geographic regions [41]. The abstract framework of a quantum multicomputer recognizes that large-scale systems demand heterogeneous interconnects [42]; in most quantum-computing technologies, it may not be possible to build monolithic systems that contain, couple, and control billions of physical qubits. Van Meter et al. [43] extended this architectural framework with a design based on nanophotonic coupling of electron-spin quantum dots that explicitly uses multiple levels of interconnect with varying coupling fidelities (resulting in varying purification requirements), as well as the ability to operate with a very low yield of functional devices. Although that proposed system has many attractive features, concerns about the difficulty of fabricating optical components of adequately high quality and the desire to reduce the page 2 lattice cycle time of the surface code led to the system design proposed in this paper.

B. Layered framework[edit]

A good architecture must have a simple structure while also efficiently managing the complex array of resources in a quantum computer. Layered architectures are a conventional approach to solving such engineering problems in many fields of information technology, and Ref. [14] presents a layered software architecture for designing quantum computers. Our architecture, which describes the physical design of the quantum computer, consists of five layers, where each layer has a prescribed set of duties to accomplish. The interface between two layers is defined by the services a lower layer provides to the one above it. To execute an operation, a layer must issue commands to the layer below and process the results. Designing a system this way ensures that related operations are grouped together and that the system organization is hierarchical. Such an approach allows quantum engineers to focus on individual challenges, while also seeing how a process fits into the overall design. By organizing the architecture in layers, we deliberately create a modular design for the quantum computer. The layered framework can be understood by a control stack composed of the five layers in the architecture. Figure 1 shows an example of the control stack for the quantum-dot architecture we propose here, but the particular interfaces between layers will vary according to the physical hardware, quantum error-correction scheme, and so on, that one chooses to implement. At the top of the control stack is the Application layer, where a quantum

FIG. 1. Layered control stack that forms the framework of a quantum-computer architecture. Vertical arrows indicate services provided to a higher layer.

algorithm is implemented and results are provided to the user. The bottom Physical layer hosts the raw physical processes supporting the quantum computer. The layers between (Virtual, Quantum Error Correction, and Logical) are essential for shaping the faulty quantum processes in the Physical layer into a system of highaccuracy, fault-tolerant [44] qubits and quantum gates at the Application layer.

C. Interaction between layers[edit]

Two layers meet at an interface, which defines how they exchange instructions or the results of those instructions. Many different commands are being executed and processed simultaneously, so we must also consider how the layers interact dynamically. For the quantum computer to function efficiently, each layer must issue instructions to layers below in a tightly defined sequence. However, a robust system must also be able to handle errors caused by faulty devices. To satisfy both criteria, a control loop must handle operations at all layers simultaneously while also processing syndrome measurement to correct errors that occur. A prototype for this control loop is shown in Fig. 2. The primary control cycle defines the dynamic behavior of the quantum computer in this architecture since all operations must interact with this loop. The principal purpose of the control cycle is to successfully implement quantum error correction. The quantum computer must operate fast enough to correct errors. Still, some control operations necessarily incur delays, so this cycle does not simply issue a single command and wait for the result before proceeding—Pipelining is essential [10,45]. A related issue is that operations in different layers occur on drastically different time scales, as discussed later in Sec. VII. Figure 2 also describes the control structure needed for the quantum computer. Processors at each layer track the current operation and issue commands to lower layers. Layers 1 through 4 interact in the loop, whereas the Application layer interfaces only with the Logical layer, since it is agnostic about the underlying design of the quantum computer (which is explained in Sec. VI).

D. The QuDOS hardware platform[edit]

The layered framework for quantum computing was developed in tandem with a specific hardware platform, known as QuDOS (quantum dots with optically controlled spins). The QuDOS platform uses electron spins within quantum dots for qubits. The quantum dots are arranged in a two-dimensional array; Fig. 3 shows a cutaway rendering of the quantum-dot array inside an optical microcavity, which facilitates control of the electron spins with laser pulses. We demonstrate that the QuDOS design is a promising candidate for large-scale page 3

FIG. 2. Primary control cycle of a quantum computer with the layered architecture. Whereas the control stack in Fig. 1 dictates the interfaces between layers, the control cycle determines the timing and sequencing of operations. The dashed box encircling the Physical layer indicates that all quantum processes happen exclusively here, and the layers above process and organize the operations of the Physical layer. The Application layer is external to the loop since it functions without any dependence on the specific quantumcomputer design.
FIG. 3. Quantum dots in a planar optical microcavity form the basis of the QuDOS hardware platform. (a) The quantum dots are arranged 1 �m apart in a two-dimensional square array. The quantum dots trap single electrons, whose spins will be used for quantum information processing. (b) Side view. The electron spins are manipulated with laser pulses sent into the optical cavity from above, and two neighboring quantum dots can be coupled by a laser optical field that overlaps them. The purple and green layers are AlGaAs and GaAs, grown by molecular-beam epitaxy. The alternating layers form a distributed-Bragg-reflector (DBR) optical cavity which is planar, confining light in the vertical direction and extending across the entire system in horizontal directions.

page 4 quantum computing, beginning with an analysis of the hardware in the Physical layer.


The essential requirements for the Physical layer are embodied by the DiVincenzo criteria [3], but we are also interested in performance of the quantum hardware. The time scale of operations and the degrees of errors, both systematic and random, are critical parameters that determine the size and speed of the computer. This section discusses the essential hardware components of a quantum computer, accompanied by the QuDOS platform we introduce as an example. We conclude by analyzing the performance of the QuDOS hardware. We caution that many of the required hardware elements are still under experimental development, but we choose those discussed below as examples to establish time scales that will affect higher layers of the architecture.

A. Physical qubit[edit]

A quantum computer must have the ability to store information between processing steps; the object fulfilling this role is conventionally known as the physical qubit. A physical qubit may be more complex than a two-level system, and this issue is addressed by Layer 2 in the architecture, where control operations are used to form a true quantum bit as an information unit (see Sec. III A). Examples of physical qubits include trapped ions, photonpolarization modes, electron spins, and quantum states in superconducting circuits [1]. The remainder of the Physical layer is devoted to controlling and measuring the physical qubit. The layered-architecture design is flexible in the sense that the Physical layer can be tailored to a specific hardware, such as superconducting circuit qubits, with minimal change to higher layers such as error correction. The physical qubit we consider in QuDOS is the spin of an electron bound within an InGaAs self-assembled quantum dot (QD) surrounded by GaAs substrate [46–51]. These QDs can be optically excited to trion states (a bound electron and exciton), which emit light with a wavelength of approximately 900 nm when they decay. A transverse magnetic field splits the spin levels into two metastable ground states [52], which will later form a two-level system for a virtual qubit in Layer 2. The energy separation of the spin states is important for two reasons related to controlling the electron spin: First, the energy splitting facilitates control with optical pulses, as explained in Sec. II C. Second, there is continuous phase rotation between spin states j "i and j #i around the z axis on the qubit Bloch sphere, which in conjunction with timed optical pulses provides complete unitary control of the electron spin vector.

B. Host system[edit]

For our purposes, the host system is the engineered environment of the physical qubit that supports computing. Examples include the trapping fields in ion-trap designs, the waveguides in optical quantum computing, and the diamond crystal surrounding nitrogen-vacancy centers [1]. The host system defines the immediate environment of the physical qubit, which will be important for characterizing noise that affects quantum operations. We noted above that, in QuDOS, the electron spin is bound within a quantum dot. These quantum dots are embedded in an optical microcavity, which will facilitate quantum-gate operations via laser pulses. To accommodate the two-dimensional array of the surface code detailed in Layer 3, this microcavity must be planar in design, so the cavity is constructed from two distributed Bragg reflector (DBR) mirrors stacked vertically with a =2 cavity layer between, as shown in Fig. 3. This cavity is grown by molecular-beam epitaxy. The QDs are embedded at the center of this cavity to maximize interaction with antinodes of the cavity field modes. Using molecular-beam epitaxy, high-quality (Q > 105) microcavities can be grown with alternating layers of GaAs=AlAs [53]. The nuclei in the quantum dot and surrounding substrate have nonzero spin, which is an important source of noise (see Sec. II F).

C. 1-qubit gate mechanism[edit]

The 1-qubit gate manipulates the state of a single physical qubit. This 1-qubit gate mechanism is still a physical process; only later will these physical control operations be combined into a ‘‘virtual gate’’ (see Sec. III B). Still, it is important that the Physical layer delivers sufficient control of the physical qubit. Full unitary control of a qubit requires at least two adjustable degrees of freedom, such as rotation around two axes on the Bloch sphere, and three freely adjustable parameters [54]. The 1-qubit operations in QuDOS are developed using a transverse magnetic field and ultrafast laser pulses [51–55]. The magnetic field provides a constant-angular-frequency rotation around the z axis, while laser pulses enact a power-dependent rotation around an orthogonal axis, which we label x. The first nonideal behavior we consider is that the laser pulse has some finite duration, so that x and z precession happen concurrently, which impairs manipulation of the spin Bloch vector. To remedy this, we introduce ‘‘Hadamard pulses’’ [56]: One tunes the laser pulse power and duration to make the pulse-driven x-axis rotation equal in angular frequency to the z-axis precession from the magnetic field, so that the axis of rotation becomes H¼ 1ffiffi 2 p ðxþzÞ. A ‘‘� pulse’’ around this axis is a Hadamard gate. By using two Hadamard pulses and rotation Rzð�Þ via free precession in the magnetic field, we can construct any x-axis rotation by Rxð�Þ ¼ H � Rzð�Þ � H, as shown in Fig. 4. By implementing Hadamard pulses, we can obtain high-fidelity operations LAYERED A page 5

FIG. 4. Hadamard pulses in QuDOS. (a) A short pulse sequence generates a x-axis rotation on the spin Bloch sphere with two Hadamard pulses and z-axis precession from the magnetic field. The duration of the pulses and the delay between them is proportional to the Larmor period, TL. (b) Bloch-sphere diagrams showing the axis of rotation at each time during the sequence.

with pulses of finite duration. A challenging problem for QuDOS is how the system executes millions of control operations in parallel. We envision an optical imaging system consisting of an array of MEMS mirrors to individually steer laser control beams toward or away from quantum dots, along with electro-optic modulators to precisely control laser pulse timing. We discuss this approach in Appendix A, but rigorously engineering such a system is beyond our scope. D. 2-qubit gate mechanism The 2-qubit operation couples two physical qubits, which can generate entanglement. This mechanism is crucial for quantum computing, yet it is often difficult to implement experimentally. For example, entangling gates like CNOT are used frequently in quantum error correction, so developing fast, high-fidelity 2-qubit gate mechanisms is imperative for large-scale quantum information processing. In many cases, the 2-qubit gate is the process that defines the speed and accuracy of a quantum computer. The construction of a practical, scalable 2-qubit gate in QuDOS remains the most challenging element of the hardware, and various methods are currently under development. A fast 2-qubit gate, fully controlled optically, would certainly be attractive, and early proposals [47] identified the importance of employing the nonlinearities of cavity QED. Ref. [47] suggests the application of two lasers for both 1-qubit and 2-qubit control; more recent developments have indicated that both 1-qubit gates [55,57,58] and 2-qubit gates [59] can be accomplished using only a single optical pulse. We consider a 2-qubit gate via the dispersive interaction proposed in Ref. [59]. The critical figure of merit for the cavity QED system is the cooperativity factor C, which is proportional to the cavity quality factor Q divided by the cavity volume V. For QuDOS, we envision transverse cavity confinement entirely due to the extended microplanar microcavity arrangement, in which cooperativity factors are enhanced by the angle dependence of the cavity response, an effect that is enlarged by high index-ofrefraction contrast in the alternating mirrors of the DBR stack [46]. While existing cooperativity factors achieved this way are not estimated to be high enough to produce quantum gates with error rates sufficiently low for faulttolerant quantum computing, advanced control techniques and multispin encodings (such as for ‘‘virtual qubits’’; see Sec. III) may enable this technology to function with acceptable error rates. Ref. [59] estimates that this gate will require 10–100 ns to execute. For the present analysis we assume the value 32 ns, which coincides with a virtual gate in Sec. III B. Further enhancements to speed or gate fidelity or both may be available by introducing exchange interactions using microcavity polaritons [60]; studying this possibility is the subject of future work. E. Measurement readout Measurement is another essential component of quantum computing. At a bare minimum, one must be able to read the final result of a calculation, but typically measurement is used extensively in fault-tolerant quantum error correction. For this reason, quantum computers may require measurement that is comparable in speed and accuracy to the control operations. Moreover, many situations call for quantum nondemolition (QND) measurement, where the physical qubit is projected into an eigenstate of the measurement operator. To illustrate a counterexample, consider a qubit defined by the ground and the first optically excited states of a quantum dot. A possible measurement scheme is to detect a photon emission, which would indicate that the qubit was in the excited state. However, the final state of the qubit is the ground state for either measurement outcome, which is destructive measurement, so this procedure cannot be repeated. Conversely, QND measurement is highly desirable because it can be repeated, so that classical readout noise can be reduced by time averaging. QuDOS requires a QND measurement scheme which is still under experimental development. The proposed mechanism (shown in Fig. 5) is based on Faraday/Kerr rotation. The underlying physical principle is as follows: An off-resonant probe pulse impinges on a quantum dot, and the optical pulse receives a different phase shift depending on whether the quantum-dot electron is in the spin-up or spin-down state. (These states are separated in energy by the external magnetic field.) Sensitive photodetectors combined with homodyne-detection measure the phase shift to enact a projective QND measurement on the electron spin. Several results in recent years have demonstrated the promise of this mechanism for measurement: page 6

FIG. 5. A dispersive quantum nondemolition (QND) readout scheme for QuDOS. (a) A probe pulse is sent into a microcavity containing a charged quantum dot. (b) The cavity-enhanced dispersive interaction between the pulse and the electron spin creates a state-dependent phase shift in the light that leaves the cavity. Measurement of the phase shift can perform projective measurement on the electron spin.

multishot experiments by Berezovsky et al. [61] and Atatu¨re et al. [62] have measured spin-dependent phase shifts in charged quantum dots, and Fushman et al. [63] observed a large phase shift induced by a neutral quantum dot in a photonic crystal cavity. Most recently, Young et al. observed a significantly enhanced phase shift from a quantum dot embedded in a micropillar cavity [64]. F. Noise sources and errors Noise and decoherence are the biggest obstacles to scalable quantum computing. In general, the noise sources that corrupt the physical qubit or degrade the fidelity of control operations should be characterized as well as possible. For the present analysis, we consider the noisy environment for an electron spin in QuDOS. The primary noise in this system is dephasing, likely caused by the inhomogeneous distribution of nuclear spins in the quantum dot. The ensemble dephasing is characterized by T� 2 � 2 ns, while the intrinsic dephasing is characterized by T2 � 3 �s [65]. When the noise experienced by a qubit is dominated by dephasing, one can counteract decoherence with control sequences tailored to this noise source [66]. Section IIIA introduces a decoupling scheme designed specifically for QuDOS. G. Hardware performance summary We summarize the execution times for the essential Layer 1 (L1) operations in QuDOS in Table I. These are the quantum processes which are the building blocks of quantum information operations in Layers 2 and above. For a complete quantum processor, however, one would also have to consider the classical control hardware and the engineering concerns, such as delays, which may occur in a large system. For example, Ref. [34] considers the implications of classical control wires, such as routing concerns, signal timing, and the generation of heat in lowtemperature devices. Although engineering of classical control hardware is an important problem, it lies outside the scope of our present analysis, and we reserve it for future work. III. LAYER 2: VIRTUAL The Virtual layer is where quantum effects in the Physical layer are first cast into information primitives— virtual qubits and quantum gates. We use ‘‘virtual’’ as it is defined in the field of computer science, where a virtual object obeys a predetermined set of behaviors, without specifying the structure of this object. As an example, a virtual qubit may be defined by a decoherence-free subspace [67–69] constructed from three electron spins; when considered as a whole, three spins have many more degrees of freedom than a single qubit. Similar behavior is seen in the quantum gates in QuDOS, which actually consist of a

TABLE I. Parameters for Layer 1 quantum operations. Spin-phase precession is determined by the spin-state energy splitting due to an external magnetic field. To implement a Hadamard gate, the broadband pulse time is 1= ffiffiffi8 p of the Larmor period (TLarmor). Times for entangling operation and QND measurement are estimated from simulation.
Operation Mechanism Duration Notes
Spin-phase precession (�z axis) Magnetic-field splitting of spin energy levels 40 ps Inhomogeneous nuclear environment causes spectral broadening in Larmor frequency, which is the source of T�2 processes.
Spin-state rotation pulse Stimulated Raman transition with broadband optical pulse 14 ps Red-detuned from the ground-state–trion transitions of the spins
Entangling operation Nonlinear phase shift of spin states via coupling to a common cavity mode 32 ns CW laser signal modulated by an electro-optic modulator (EOM)
QND measurement Dispersive phase shift of light reflected from planar cavity 1 ns CW laser signal modulated by an EOM

page 7

FIG. 6. The mechanics of the Virtual layer. The outputs of Layer 1 are combined in controlled sequences to produce virtual qubits and gates. Arrows indicate how the output of one process is used by another process.

sequence of laser pulses. This transcription process of converting many physical elements into a virtual information unit is the task of Layer 2, and we clarify the functions of this layer in this section. Figure 6 gives an overview of the Virtual layer processes in QuDOS. In a general sense, the Virtual layer makes the Physical layer robust to systematic errors. This effect is seen in both virtual qubits and gates, where we enforce symmetries in the system (by careful design of control operations) which cause correlated errors to cancel by interference. The simplest example of this behavior is the Hahn spin-echo sequence [70], and in fact decoupling techniques play a prominent role in how we construct a virtual qubit.

A. Virtual qubit[edit]

The virtual qubit shapes the underlying physical qubit into a two-level system which approximates an ideal qubit. However, the virtual qubit is modeled as having some finite amount of decoherence, such as the depolarizing channel [54]. Where applicable, dynamical decoupling [71–73] and/or decoherence-free subspaces [67–69] are used to create long-lived virtual qubits, and the residual decoherence characterizes the lifetime of the virtual qubit. In what follows, we consider how to construct a virtual qubit with a charged quantum dot, including the mitigation of several nonideal effects in this system. In QuDOS, the virtual qubit is created from the two metastable spin states of an electron confined to a QD. As discussed in Sec. II F, the raw physical system has dephasing time T� 2 � 2 ns [65] caused by an inhomogeneous distribution of nuclear spins in the environment of the electron. This dephasing time is insufficient for quantum error correction in Layer 3, so this system must be augmented with dynamical decoupling techniques [74,75], which extend the dephasing time of the virtual qubit into the microsecond regime [65]. Constructing the virtual qubit in QuDOS requires Layer 2 to conceal the complexity of controlling the QD spin state. Because the physical-qubit Bloch vector continuously rotates around the �z axis, control pulses must be accurately timed so that they perform the desired operation. Furthermore, control of the QD spin is complicated by the inhomogeneous nuclear-spin environment which causes the �z-axis rotation to proceed at a somewhat uncertain angular frequency. This problem is mitigated by a dynamical decoupling (DD) sequence, so that the system is decoupled from environmental noise and brought into a precisely controlled reference frame at a predictable time. Figure 7(a) illustrates the ‘‘8H’’ decoupling sequence (so named because it uses eight Hadamard pulses), which is appropriate for use in QuDOS. This control sequence is designed both to decouple a qubit from dephasing noise and to compensate for systematic pulse errors in the presence of a strong but slowly fluctuating drift term in the qubit Hamiltonian, which is the case for optically controlled quantum dots in a strong magnetic field. Although longer sequences consisting of more pulses may in theory decouple to higher fidelity, we have chosen a sequence of just eight Hadamard pulses to minimize execution time. Instead of using a more common sequence like the Carr-Purcell (CP) sequence [76,77] or Uhrig dynamical decoupling (UDD) [78], the sequence in Fig. 7 is custom designed to eliminate to first order the errors that occur in both the free evolution and the control of the virtual qubit. (CP and UDD cannot accomplish the latter.) We note, however, that the 8H sequence does have a structure similar to the CP sequence. Figure 8 shows the simulated effectiveness of 8H as compared to CP and UDD. We have selected � ¼ 1 ns [from Fig. 7(a)], so one iteration of the sequence requires 8 ns. Because this sequence is specifically designed to account for the errors particular to QuDOS, the performance page 8

FIG. 7. A special dynamical decoupling sequence for QuDOS, known as 8H since it requires eight Hadamard pulses. TL is the Larmor period determined by the external magnetic field (see Table I). (a) Timing specification for the 8H sequence, where � is an arbitrary time. Each of the pulse pairs enacts a � rotation around the �x axis of the virtual-qubit Bloch sphere, as shown in Fig. 4. For 8H to work efficiently, � � T2. (b) Four 8H sequences in a row interleaved with arbitrary gates formed from three Hadamard pulses (orange). The overall sequence forms a virtual gate by way of a BB1 compensation sequence.

exceeds that of the more common dynamical-decoupling schemes. Nonetheless, 8H may be very effective in other quantum information systems where the physical qubit states are separated in energy and the control pulses have a duration that is comparable to the free-precession (e.g., Larmor) period of the qubit Bloch vector.

FIG. 8. Simulation of the decoupling effectiveness of the 8H sequence compared to CP and UDD (each using 4 X gates) in the presence of dephasing noise and control errors. Here, ‘‘pulse error’’ is a systematic, relative deviation in the energy of every pulse. In all cases, two Hadamard pulses are combined to produce an approximate X gate, as in Fig. 4. The vertical axis is infidelity after evolution of the sequence in Fig. 7(a) with � ¼ 1 ns; here, infidelity is 1 � F ¼ 1 � �II, where �II is the identity-to-identity matrix element in process tomography for the decoupling gate sequence with random noise. Since we aim to execute virtual gates with 1 �F<10�3, laser-pulse errors must be less than 1% in order for the virtual-qubit memory error rate to be adequately low.

B. Virtual gate[edit]

Virtual gates manipulate the state of the virtual qubit by combining physical control operations in Layer 1 in a manner that creates destructive interference of control errors. Quantum operations must be implemented by physical hardware, which is ultimately faulty to some extent. Many errors are systematic, so that they are correlated in time, even if they are unknown to the quantumcomputer designer. Virtual gates suppress systematic errors as much as possible in order to satisfy the demands of the error-correction system in Layer 3. Efficient schemes exist for eliminating systematic errors. Compensation sequences can correct correlated errors in the gate operations in Layer 1 [79,80]. This situation arises often for errors due to imperfections in the control operations, such as fluctuations in laser intensity or the strength of the coupling of a quantum-dot electron to an optical field (caused by fabrication imperfections). If these errors are correlated on time scales longer than operations in this architecture, a compensation sequence is effective for generating a virtual gate with lower net error than each of the constituent gates in the sequence. Many compensation sequences are quite general, so that error reduction works without knowledge of the type or magnitude of error. Dynamically corrected gates are an alternative scheme where one tunes the time-dependent Hamiltonian of the control operations [81]. Beyond such open-loop control techniques, it is also desirable to characterize the accuracy of operations in the Virtual layer, especially multiqubit gates and entangled states. Systematically evaluating quantum operations is an important component of a research program to develop quantum computers and merits further investigation; however, it is beyond our present scope. page 9 In QuDOS, the ultrafast pulses in Layer 1 would ideally induce a state rotation in the spin basis (two-level system), but inevitably the physical system will suffer from some loss of fidelity by both systematic and random processes. We attempt to cause destructive interference of any systematic errors—from both the environment and control pulses—by embedding a BB1 compensation sequence within a train of 8H dynamical decoupling sequences, as shown in Fig. 7(b). The BB1 sequence combines four pulses with an unknown, systematic bias in a manner that reproduces the action of a single pulse that has much higher fidelity. This approach is motivated by the properties of the physical qubit. The electron spin has a strong but slowly fluctuating drift term in its Hamiltonian because of the magnetic field and the nuclear-spin environment. The 8H sequence brings the qubit ‘‘into focus’’ (analogous to a ‘‘spin echo’’) only at prescribed instants, which are when the BB1 pulses are applied. This approach is more accurate than a BB1 sequence without refocusing because of the time required to implement rotations on the physical qubit Bloch sphere using Hadamard pulses, for the same reasons that 8H is more effective at decoupling than CP or UDD sequences, as shown in Fig. 8. The BB1 compensation sequence requires four arbitrary gates [79]; hence, the virtual gate with error cancellation requires 32 ns.

C. Measurement of virtual qubits[edit]

Measurement is a crucial operation that must also be applied to the virtual qubit in a manner consistent with other control processes. For example, dynamical decoupling prevents measurement by isolating a qubit from environment interactions, so DD may have to be suspended during readout. Since measurement plays a crucial role in error correction, this mechanism should be made as fast and efficient as possible, and a slow measurement process may suffer loss of fidelity if the physical qubit decoheres quickly without DD. Even if the measurement process is much faster than qubit decoherence, classical noise in the measurementreadout signal could be a concern. If the Physical layer provides QND measurement, then the Virtual layer can repeat the measurement of a virtual qubit multiple times and overcome noise in readout circuitry by a majority poll of discrete measurement outcomes. This is a simple yet robust way to suppress measurement errors. For example, if the optical measurement pulse in QuDOS requires 1 ns, then measurement could be repeated about 30 times in the same window of time as a virtual gate. Another possibility is to couple a virtual qubit to one or more ancilla qubits that facilitate measurement [82]. In such a scheme, the measurement process at the Physical layer could be destructive, but since only the ancilla is destroyed, the backaction on the original qubit is QND measurement, which can be repeated. Measurement of the virtual qubit in QuDOS requires that the DD sequence be halted, because the 8H sequence interferes with readout. Since the measurement pulse is in the �z basis, rotations around the �z axis from the magnetic environment do not affect the outcome. Neglecting DD during measurement is acceptable because the longitudinal (T1) relaxation time is very long compared with the measurement pulse duration [83,84].


Fault-tolerant quantum error correction (QEC) is essential for large-scale quantum computing. In Sec. VI B we analyze an implementation of Shor’s factoring algorithm that requires an error-per-gate of order 10�15, which is simply infeasible on faulty hardware, even using Layer-2 techniques like dynamical decoupling. The action of error correction on a quantum information system is to pump entropy out in the form of an error syndrome; in the process, new resources—logical qubits and gates—are created. Whereas Layer 2 causes correlated errors to cancel, Layer 3 isolates and removes arbitrary errors, so long as the error rate is below a threshold [85]. If this condition is met, QEC can in principle produce arbitrarily low-error logical qubits and gates. Such complete error suppression is necessary because quantum algorithms in the Application layer assume that logical qubits and gates are error free. The field of quantum error correction has become too broad to cover in its entirety [54,86,87]. Instead, we analyze the case of stabilizer codes [88], and we specifically consider the surface code [2,89,90] for QuDOS. We select the surface code for its high threshold and two-dimensional nearest-neighbor interaction geometry. This section focuses on the aspects of quantum error correction that are relevant for a quantum-computer architecture, such as determining the size of a code sufficient for a certain application, as well as how the errors are tracked by Pauli frames in classical hardware. Figure 9 shows the surface-code operations in Layer 3 for QuDOS. Other error-correction schemes besides the surface code could also be implemented in a layered architecture. As examples, the C4=C6 code [91,92] and Bacon-Shor codes [32–34,93] have also received significant attention as viable schemes for fault-tolerant quantum computation. Because the layered architecture is modular, replacing the surface code with another QEC scheme is possible as long as the Virtual layer supports the necessary operations of the new code. In general, the QEC code chosen is likely to impact many aspects of a quantum computing system, such as device geometry, connectivity, and sensitivity to defective components, so that the structure and behavior of the computer is defined in large part by the selected code. For this reason, much attention should be devoted to optimizing Layer 3 in any quantum-computer architecture. One key message is worth emphasizing. The threshold error rate of an error-correcting code is defined as the error rate at which error correction begins to show a net gain in protecting information. A functioning quantum page 10

FIG. 9. Process translation in Layer 3 in QuDOS. A surface code is constructed with virtual qubits and gates, ultimately yielding logical qubits and operations. The arrows in yellow along the bottom are outputs of Layer 2, whereas the green arrows at the top are the outputs of Layer 3. Dashed black arrows indicate that the output of one process is used by another process.

error-correction system must operate below threshold, and a practical system must operate well below threshold. We show in this section that the resources required for error correction become manageable when the hardware error rate is about an order of magnitude below the threshold error rate of the chosen code.

A. Estimating the strength of error correction needed[edit]

We consider how to estimate the degree of error correction required for a given application because this determines the necessary amount of resources in the computer. Quantum error-correction schemes generate protected code spaces within a larger Hilbert space formed from many qubits. The tradeoff for reducing logical errors is that, instead of requiring a single qubit, the quantum computer now requires many virtual qubits to produce a logical qubit. The number of virtual qubits required for a single logical qubit is an important resource-usage quantity. It depends on the performance aspects of the quantum computer: (i) error per virtual gate ("V), which is an input to Layer 3 from Layer 2, (ii) threshold error per virtual gate of the errorcorrecting code ("thresh), (iii) distance (d) of the code, (iv) maximum error per logical gate ("L), which is bounded on the upper end by the performance requirements of the quantum algorithm in Layer 5. To determine "L, the simplest approach, the KQ product, assumes the worst case. If the quantum algorithm has a circuit with logical depth K acting on Q logical qubits, then the maximum failure probability is given by Pfail ¼ 1 � ð1 � "LÞKQ � KQ"L (1) for small "L. Therefore, we demand that "L � 1=KQ. Given these quantities, the average error per logical gate in a code operating well below threshold may be closely approximated [54,85,86,94–96] by "L � C1 � C2 "V "thresh �ðdþ1Þ=2


where C1 is a constant determined by the specific implementation of the code, C2 � 1, and, by assumption, "V � "thresh. The data in Ref. [96] suggests C1 � 0:13 and C2 � 0:61 for the surface code, which we now use as an example. Given a known "V and code-specific quantities f"thresh; C1; C2g, one can determine the necessary distance d such that the probability of failure of an entire quantum algorithm is sufficiently small. For comparison, Aliferis presents similar analysis for concatenated codes such as the Bacon-Shor code [97]. Equation (2) illustrates that the error per virtual gate should be "V < 0:2"thresh; otherwise, the code distance, and hence the size of the quantum computer, will be impractically large. Table II provides an example of these

TABLE II. Parameters determining the size of the surface code in QuDOS for an implementation of Shor’s factoring algorithm.
Parameter Symbol Value
Threshold error per virtual gate [96] "thresh 9 � 10�3
Error per virtual gate "V 1 � 10�3
Circuit depth (lattice refresh cycles) K 1:6 � 1011
Logical qubits (Shor, Sec. VI B) Q 72 708
Error per lattice refresh cycle "L 2:6 � 10�20
Surface-code distance d 31
Virtual qubits per logical qubit nV=nL 6240

page 11 calculations for the QuDOS quantum computer. Error per virtual gate ("V) is also assumed, and the K and Q values are for Shor’s algorithm factoring a 1024-bit integer (see Sec. VI B). We require that "L � 10�2=KQ, so that the logical error probability of the quantum algorithm is less than 1%. Determining the necessary strength of error correction also indicates how large the quantum computer is in terms of qubits.We can estimate the number of virtual qubits per logical qubit, or nV=nL, by considering the minimum area needed for the two lattice defects, which make up a logical qubit, separated by distance d in the surface code [2]. For a typical set of parameters, as might be required in a large-scale computing application such as Shor’s factoring algorithm [98], 6240 virtual qubits are needed to construct a logical qubit. This is a nontrivial overhead, because quantum algorithms require a substantial number of logical qubits, as we discuss in greater detail in subsequent sections. For example, quantum simulation algorithms may require about 1000–10 000 logical qubits [99,100] and integer factoring may require 100 000 logical qubits or more, depending on the methods of calculating arithmetic [101]. More detail on why so many logical qubits are necessary is given in Sec. VB. Combining the size of quantum computations with the requirements of error correction means that large-scale quantum-computing architectures will require millions or billions of virtual qubits (and hence physical qubits).

B. Pauli frames[edit]

A Pauli frame [91,102] is a simple and efficient classical computing technique to track the result of applying a series of Pauli gates (X, Y, or Z) to single qubits. The Gottesman- Knill theorem implies that tracking Pauli gates can be done efficiently on a classical computer [103]. Many quantum error-correction codes, such as the surface code, project the encoded state into a perturbed code word with erroneous single-qubit Pauli gates applied (relative to states within the code space). The syndrome, which is the set of measurements that identifies the most likely error configuration, reveals what these Pauli errors are, up to undetectable stabilizers and logical operators, and error correction is achieved by applying those same Pauli gates to the appropriate qubits (since Pauli gates are Hermitian and unitary). However, quantum gates are faulty, and applying additional gates may introduce more errors into our system. Rather than applying every correction operation, one can keep track of what Pauli correction operation would be applied, and continue with the computation. This is possible because the operations needed for error correction are in the Clifford group. When a measurement in a Pauli X, Y, or Z basis is finally made on a qubit, the result is modified based on the corresponding Pauli gate that should have been applied earlier, as in Fig. 10. This stored Pauli gate is called the Pauli frame [91,102], since, instead of applying a Pauli gate, the quantum computer changes the reference

FIG. 10. Example of a Pauli frame evolving over time with entries corresponding to virtual qubits forming a surface code. Each horizontal slice is the Pauli frame at that time. For example, if the qubit in the top-front-corner position is measured in the X basis, the interpreted result is the negation of the observed outcome, because the Pauli frame Z anticommutes with this measurement basis.

frame for the qubit, which can be understood by remapping the axes on the Bloch sphere, rather than moving the Bloch vector. The Pauli frame is maintained as follows. Denote the Pauli frame at time t as Ft: Ft ¼ O j PtðjÞ; (3) where PtðjÞ ¼ fI; X; Y; Zg is an element from the Pauli group corresponding to qubit j at time t. Any Pauli gate in the quantum circuit is multiplied into the Pauli frame and is not implemented in hardware, so Ftþ1 ¼ ð N jUfI;X;Y;ZgÞFt for all Pauli gates UfI;X;Y;Zg in the circuit at time t. Other gates UC in the Clifford group are implemented, but they will transform the Pauli frame by Ftþ1 ¼ UCFtUy C: (4) The quantum-computer operations proceed normally, with the only change being how the final measurement of that qubit is interpreted. The set of Clifford gates is sufficient for Layer 3, although the next section describes another Pauli frame for non-Clifford logical operations. We emphasize that the Pauli frame is a classical object stored in the digital circuitry that handles error correction. Pauli frames are nonetheless very important to the functioning of a surface-code quantum computer. Layer 3 uses a Pauli frame with an entry for each virtual qubit in the error-correcting code. As errors occur, the syndrome processing step identifies a most-likely pattern of Pauli errors. Instead of applying the recovery step directly, the Pauli frame is updated in classical memory. The Pauli gates form a closed group under multiplication (and the global phase of the quantum state is unimportant), so the Pauli frame tracks only one of four values (X, Y, Z, or I) for each virtual qubit in the lattice.


The Logical layer takes the fault-tolerant resources from Layer 3 and creates a logical substrate for universal page 12

FIG. 11. Organization of processes in the Logical layer. Logical qubits from Layer 3 are unaltered, but faulty singular states are distilled into high-fidelity states jYi ¼ 1ffiffi 2 p ðj0i þ ij1iÞ and jAi ¼ 1ffiffi 2 p ðj0i þ ei�=4j1iÞ. The distilled states are used to create arbitrary gates with specialized quantum circuits [39,105–107].

quantum computing. This task requires additional processing of error-corrected gates and qubits to produce any arbitrary gate required in the Application layer, as shown in Fig. 11. Quantum error correction provides only a limited set of gates—To see why, consider that no finite number of syndrome bits can distinguish arbitrarily small rotation gate errors. A common set of gates provided by QEC is the Clifford group; although circuits from this set can be simulated efficiently on a classical computer by the Gottesman-Knill theorem [54], the Clifford group forms the backbone of quantum circuits. Still, some QEC schemes, such as the surface code, do not provide the full Clifford groupwithout some sort of ancilla.We identify the set of fault-tolerant gates generated by Layer 3 without the use of ancillas as the fundamental gates. The Logical layer then constructs arbitrary gates fromcircuits of fundamental gates and ancillas injected into the error-correcting code. For example, surface code architectures inject and purify the ancillas jYi¼ 1ffiffi 2 p ðj0iþij1iÞ and jAi¼ 1ffiffi 2 p ðj0iþei�=4j1iÞ; then the surface code consumes these ancillas in quantum circuits to produce S ¼ eið�=4Þ�Z and T ¼ eið�=8Þ�Z gates, respectively [2,54]. This section discusses the important functions of the Logical layer: implementing logical Pauli frames; distilling ancilla states like jYi and jAi; implementing the full Clifford group in the surface code without measurement; and approximating arbitrary quantum gates for the Application layer.

A. Fundamental gates and the logical Pauli frame[edit]

Fundamental gates are provided natively by the errorcorrecting code in Layer 3. For example, Table III shows the fundamental gates used in QuDOS. In practice, Pauli gates are implemented with a logical Pauli frame, which is qualitatively the same as the Pauli frame in Layer 3 for virtual qubits (Sec. IVB). However, in Layer 4 we may also need to apply gates UNC outside the Clifford group (NC). The gate we actually implement, U0 NC, results from a Pauli frame transform: U0 NC ¼ FtUNCFy t : (5) Note the distinction between this expression and Eq. (4), where the Pauli frame is changed by a Clifford gate. The fundamental gate set in Table III is particular to the surface code, and it is not the full Clifford group because it is missing the phase gate S. Section VC illustrates a method to construct S using an ancilla, without measurement, which is efficient since it uses a small number of fundamental gates and the ancilla can be reused. The remaining logical gates to produce a universal set in the

TABLE III. Fundamental gates in QuDOS using surface-code QEC (Ref. [2]). The execution time here is one possible implementation, but in many cases the surface-code computation can be deformed into other topologically equivalent circuits which yield faster execution at the expense of more spatial resources, or vice versa.
Gate Implementation Execution time [lattice steps]
X, Y, Z Pauli frame Instantaneous
CNOT Defect braiding 13dd=4e
H (Hadamard) Shift lattice 13dd=8e
MX, MZ (measurement) Measure stabilizers 1

page 13 surface code require ancilla states which are injected and distilled [2].

B. Magic-state distillation[edit]

The conventional method for making a universal set of quantum gates in a fault-tolerant manner is to produce a certain ancilla state and use it in a quantum circuit equivalent to the desired logical gate [44,54,97]. In some cases these circuits require measurement that consumes the ancilla, so that the number of ancilla states required is proportional to the number of gates in the quantum algorithm. For example, the algorithms discussed in Sec. VI require around 1012 or more ancilla states, which are typically manufactured as needed. To complicate matters, ancillas must be produced by methods that are not fault tolerant, such as initializing a virtual qubit and applying the appropriate virtual gate. This ancilla state can then be injected into a QEC code in Layer 3 [2], but it carries with it the errors in its production. Fortunately, a few ‘‘magic states’’ can be distilled by using several low-fidelity ancillas and fundamental gates to produce one high-fidelity ancilla. Once the ancilla fidelity is higher than the necessary logical-gate fidelity, we may construct arbitrary fault-tolerant logical gates.We examine here the resource costs for this process; each distillation is expensive, and very many ancillas must be distilled. We characterize the performance of the magic-state distillation because it will probably dominate the resource costs of any quantum computer that uses it. Accordingly, this is an important area for future optimizations. We focus first on distilling the ancilla state jAi ¼ 1ffiffi 2 p � ðj0i þ ei�=4j1iÞ, which is used to construct the T or �=8 phase gate [2,90,104]. In the next section, Fig. 14 provides an illustration of why this process is important by showing the fault-tolerant construction of a Toffoli gate in Layer 5 using resources in Layer 4; specifically, ancilla distillation circuits constitute over 90% of the computing effort for a single Toffoli gate. As a result, the analysis in Appendix B 1 contends that these distillation circuits account for the majority of resources in a surface-code quantum computer executing Shor’s algorithm. In particular, for every qubit used by the algorithm, approximately 10 qubits are working in the background to generate the necessary distilled ancillas. The ancilla distillation circuit in Fig. 14 shows one level of jAi distillation, but a lengthy program like Shor’s will typically require two levels (one concatenated on another). Moreover, since perhaps trillions of distilled jAi ancillas will be needed for the algorithm, we create a ‘‘distillation factory’’ [9,43], which is a dedicated region of the computer that continually produces these states as fast as possible. Speed is important, because ancilla distillation can be the rate-limiting step in quantum circuits [10]. Each jAi distillation circuit will require 15 lower-level jAi states, but they are not all used at the same time. For simplicity we will use a clock cycle for each gate equal to

TABLE IV. Resource analysis for a distillation factory. These factories are crucial to quantum computers that require ancillas for universal gates. Magic-state distillation uses Clifford gates and measurement, so the circuit can be deformed to reduce depth and increase area, or vice versa, while keeping volume approximately constant.
Parameter Symbol Value
Circuit depth 6 clock cycles
Circuit area Adistil 12 logical qubits
Circuit volume VðjAð1ÞiÞ 72 qubits � cycles
Factory rate (level n) RfactoryðjAðnÞiÞ Afactory=VðjAðnÞiÞ ancillas=cycle

the time to implement a logical CNOT (see Sec. VII for more on this point), so that, with initialization and measurement, the distillation circuit requires 6 cycles. By using jAi ancillas only when they are needed, the circuit can be compacted to require at most 12 logical qubits at a given instant. We characterize the computing effort by a ‘‘circuit volume,’’ which is the product of logical memory space (i.e., area of the computer) and time. The circuit volume of jAi distillation is VðjAð1ÞiÞ ¼ ð12 logical qubitsÞ � ð6 clock cyclesÞ ¼ 72. A two-level distillation will require 16 distillation circuits, or a circuit volume of VðjAð2ÞiÞ ¼ 1152. An efficient distillation factory with area Afactory will produce on average Afactory=VðjAð2ÞiÞ distilled ancillas per clock cycle. Analysis of this problem in the context of Shor’s algorithm is given in Appendix B 1, and Table IV lists a summary of these results.

C. Logical phase gate without measurement[edit]

The S gate, or phase gate, is the final component of the Clifford group absent from the fundamental set of faulttolerant gates discussed in Sec. VA. Previous implementations of the surface code presented a method for creating this gate by consuming a distilled jYi state in a projective measurement-based circuit [2,90]. However, this approach forces the quantum computer to distill a high-fidelity jYi ancilla for each S gate, which can be very costly in both fundamental gates and qubits. We consider an alternative method that uses the jYi ancilla without consuming it to make the S gate, which was originally presented in Ref. [97]. The circuit uses only four fundamental gates and, unlike the previous technique, is deterministic because measurement is not needed. Since jYi ancillas are not consumed, one can distill a handful of such states when a quantum computer is turned on, and then preserve them for later use. The circuit in Fig. 12 is equivalent to a simple S gate on the control qubit. This equivalence is because jYi is the þi eigenstate of the operator iY, and so the controlled-iY gate will impart a phase þi only if the control qubit is in the state j1i, which is identical to the S gate. Note also that Sy can be created page 14

FIG. 12. Circuit decomposition for a logical S gate that uses an ancilla jYi but does not consume it. The inverse operation Sy can be created by running this circuit in reverse.

by running the circuit in Fig. 12 backward. This technique allows one to implement the entire Clifford group without measurement in the surface code. Moreover, since S gates are used frequently in quantum algorithms, this improved gate construction substantially reduces the complexity of a quantum computer because fewer of the resource-intensive state distillations are necessary.

D. Approximating arbitrary logical gates[edit]

The primary function of the Logical layer is to decompose arbitrary unitary gates from the quantum algorithm into circuits that contain fundamental gates available from the QEC layer. The circuits in the Logical layer act on application qubits (used explicitly by the quantum algorithm) and ancilla logical qubits, both of which facilitate universal quantum computation, as shown in Fig. 13. Since arbitrary quantum gates are not available directly, they must be approximated in some fashion, where the total resources required is a function of the approximation accuracy. We cover briefly some of the methods that can

FIG. 13. Constructing an arbitrary gate in the Logical layer. Application qubits are visible to the quantum algorithm, while logical ancilla qubits facilitate universal quantum computation. Some ancillas are reused, such as jYi for S gates, while other ancillas are consumed, such as jAi for T gates. Often, when an ancilla is consumed in a circuit that uses measurement, the circuit is probabilistic, and the Pauli frame is updated conditional on the measurement result.

be employed to produce such arbitrary gates; a more comprehensive survey of techniques is given in Ref. [105]. When constructing approximations to a unitary operation in the Logical layer, one seeks to implement a quantum circuit that approximates the desired unitary operation with minimal overhead in terms of gates and ancillas produced by Layer 3. We denote approximation accuracy as "approx ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d � jtrðUyUapproxÞj d s


where Uapprox is the fault-tolerant quantum circuit that approximates the desired unitary U, and d is the dimensionality of these operators [106]. Several techniques exist for approximating arbitrary single-qubit gates, which can be generalized to arbitrary multiqubit gates: (i) Gate approximation sequences, such as those produced by the Solovay-Kitaev algorithm [54,107] or Fowler’s algorithm [106], generate a sequence of gates from the fault-tolerant set (e.g., the set fX; Y; Z; H; S; Tg) that approximates the desired unitary U. The depth of these sequences scales as O½logcð"approxÞ with c � 4 for Solovay-Kitaev sequences and O½logð"approxÞ for Fowler sequences. (ii) Phase kickback uses a special ancilla register and a quantum adder to produce fault-tolerant phase rotations [39,108,109]. The depth of phase-kickback circuits is O½logð"approxÞ or O½loglogð"approxÞ , depending on the quantum adder [110–112]. The ancilla register, which is not consumed and can be reused, has size m qubits to approximate a phase rotation to a precision of � 2m radians, which is also O½logð"approxÞ . (iii) ‘‘Teleportation gates’’ [29] can yield very fast quantum circuits, but typically a special-purpose ancilla required for each such gate must be computed in advance, which demands a larger and more complex quantum computer. Teleportation gates that increase performance in large-scale quantum computing are used extensively in the architecture of Ref. [10] and in the simulation algorithms analyzed in Ref. [105]. Choosing among these methods depends on the capabilities of the quantum architecture, such as available logical qubits for parallel computation, and on the performance characteristics desired for the computer.


The Application layer is where quantum algorithms are executed. The efforts of Layers 1 through 4 have produced a computing substrate that supplies any arbitrary gate needed. The Application layer is therefore not concerned with the implementation details of the quantum computer—It is an ideal quantum programming environment. We do not introduce any new algorithmic methods here, but rather we page 15 are interested in how to accurately estimate the quantum computing resources required for a target application. This analysis can indicate the feasibility of a proposed quantumcomputer design, which is a worthwhile consideration when evaluating the long-term prospects of a quantumcomputing research program. A quantum engineer could start here in Layer 5 with a specific application in mind and work down the layers to determine the system design necessary to achieve desired functionality. We take this approach for QuDOS by examining two interesting quantum algorithms: Shor’s factoring algorithm and simulation of quantum chemistry. A rigorous system design is beyond the scope of the present work, but we consider the computing resources required for each application in sufficient detail that one may gauge the engineering effort necessary to design a quantum computer based on QuDOS technology.

A. Elements of the Application layer[edit]

The Application layer is composed of application qubits and gates that act on the qubits. Application qubits are logical qubits used explicitly by a quantum algorithm (see Fig. 13). As discussed in Sec. V, many logical qubits are also used to distill ancilla states necessary to produce a

FIG. 14. AToffoli gate (jx; y; zi ! jx; y; z xyi) at the Application layer is constructed with assistance from the Logical layer, using the decomposition in Ref. [54]. There are only three application qubits, but substantially more logical qubits are needed for distillation circuits in Layer 4. The jAð2Þi ancillas are the result of two levels of distillation (jAð0Þi is an injected state) on the ancilla required for T gates. Note that, each time an ancilla is used with measurement, the Pauli frame may need to be updated. The ancilla-based circuit for S gates (see Fig. 12) is not shown here, for clarity.

universal set of gates, but these distillation logical qubits are not visible to the algorithm in Layer 5. When an analysis of a quantum algorithm quotes a number of qubits without reference to fault-tolerant error correction, often this means the number of application qubits [99,113–115]. Similarly, Application-layer gates are equivalent in most respects to logical gates; the distinction is made according to what resources are visible to the algorithm or deliberately hidden in the machinery of the Logical layer, which affords some discretion to the computer designer. A quantum algorithm could request any arbitrary gate in Layer 5, but not all quantum gates are equal in terms of resource costs. We saw in Sec. VB that distilling jAi ancillas, which are needed for T gates, is a very expensive process. For example, Fig. 14 shows how Layers 4 and 5 coordinate to produce an Application-layer Toffoli gate, illustrating the extent to which ancilla distillation consumes resources in the computer. When ancilla preparation is included, T gates can account for over 90% of the circuit complexity in a fault-tolerant quantum algorithm (cf. Ref. [10] as well). For this reason, we count resources for applications in terms of Toffoli gates. This is a natural choice, because the level of ancilla distillation, number of virtual qubits, etc., depend on the choice of hardware, page 16 error correction, and many other design-specific parameters. By comparison, the number of Toffoli gates is machine independent since this quantity depends only on the algorithm (much like the number of application qubits mentioned above). To determine error correction or hardware resources for a given algorithm, one can take the Layer-5 resource estimates and work down through Layers 4 to 1, which is an example of modularity in this architecture framework. Using the analysis in the preceding sections, an Application-layer Toffoli gate in QuDOS has an execution time of 930 �s (31 logical gate cycles including the S gate circuits, discussed in Sec. VII).

B. Shor’s algorithm[edit]

Perhaps the most well-known application of quantum computers is Shor’s algorithm, which decomposes an integer into its prime factors [98]. Solving the factoring problem efficiently would compromise the RSA cryptosystem (named after the authors of Ref. [116]). Because of the prominence of Shor’s algorithm in the field of largescale, fault-tolerant quantum computing, we analyze the resources required to factor a number of the size typical for RSA cryptography. A common key length for RSA public-key cryptography is 1024 bits. Factoring a number this large is not trivial, even on a quantum computer, as the following analysis shows. Figure 15 shows the expected run time on QuDOS

FIG. 15. Execution time for Shor’s algorithm, using the same circuit implementation as in Ref. [43]. The vertical axis shows circuit depth, in terms of Toffoli gates, and the plot is labeled with estimated runtime on the QuDOS architecture. The blue trace (with triangles) is a quantum computer whose size in logical qubits scales as necessary to compute at the speed of data (no delays). The green trace (with squares) is a machine with 105 logical qubits, which experiences rapidly increasing delays as problem size increases beyond 2048 bits since insufficient resources are available to distill ancillas for T gates, a necessary component of Shor’s algorithm. The inset shows the same data on a linear vertical scale, illustrating when the quantum computer experiences delays for lack of enough qubits.

for one iteration of Shor’s algorithm versus key length in bits for two different quantum computers: one where system size increases with the problem size, and one where the system size is limited to 105 logical qubits (including application qubits). For the fixed-size quantum computer, the runtime begins to grow faster than the minimal circuit depth when factoring numbers 2048 bits and higher. Fixing the machine size highlights the importance of the ancilla distillation factories. For this instance of Shor’s algorithm, about 90% of the machine should be devoted to distillation; if insufficient resources are devoted to distillation, performance of the factoring algorithm plummets. For example, the 4096-bit factorization devotes about 75% of the machine to distillation, but about 3 times as many factories would be needed to achieve maximum execution speed in the lower trace in Fig. 15. Many design parameters in an implementation of Shor’s algorithm can be tuned as desired; we collect the details of our analysis in Appendix B 1. We should also mention here that Shor’s algorithm is probabilistic, so a few iterations may be required [98].

C. Quantum simulation[edit]

Quantum computers were inspired by the problem that simulating quantum systems on a classical computer is fundamentally difficult. Feynman postulated that one quantum system could simulate another much more efficiently than a classical processor, and he proposed a quantum processor to perform this task [117]. Quantum simulation is one of the few known quantum algorithms that solves a useful problem believed to be intractable on classical computers, so we analyze the resource requirements for quantum simulation in the quantum architecture we propose. We specifically consider fault-tolerant quantum simulation. Other methods of simulation are under investigation [118–120], but they lie outside the scope of this work. The particular example we select is simulating the Schro¨dinger equation for time-independent Hamiltonians in firstquantized form, where each Hamiltonian represents the electron-nuclear configuration in a molecule [100,121]. An application of such a simulation is to determine groundand excited-state energy levels in a molecule. We select first-quantized instead of second-quantized form for better resource scaling at large problem sizes [122]. Figure 16 shows the time necessary to execute the simulation algorithm for determining an energy eigenstate on the QuDOS computer as a function of the size of the simulation problem, expressed in number of electrons and nuclei. First-quantized form stores the position-basis information for an electron wave function in a quantum register, and the complete Hamiltonian is a function of one- and two-body interactions between these registers. For these reasons, this method does not depend on the particular molecular structure or arrangement; hence, the method is page 17

FIG. 16. Execution time for simulation of a molecular Hamiltonian in first-quantized form, as a function of problem size. The horizontal axis is number of particles being simulated, and the plot is labeled with some interesting examples from chemistry. The vertical axis is circuit depth in Toffoli gates, and the plot is labeled with estimated runtime on QuDOS. Each simulation uses 12-bit spatial precision in the wave function and 210 time steps for 10-bit precision in readout, or at most about 3 significant figures. The linear scaling in algorithm runtime versus problem size is due to two-body potential-energy calculations, which constitute the majority of the quantum circuit. The number of potential-energy calculations increases quadratically with problem size, but through parallel computation they require linear execution time, as described in Appendix B 2.

very general. Note that the calculation time scales linearly in problem size, as opposed to the exponential scaling seen in classical methods. The precision of the simulation scales with the number of time steps simulated [99]; this example uses 210 time steps for a maximum precision of about 3 significant figures. Details of this simulation algorithm can be found in Appendix B 2.

D. Large-scale quantum computing[edit]

The factoring algorithm and quantum simulation represent interesting applications of large-scale quantum computing; for each the computing resources required of a layered architecture based on QuDOS are listed in Table V. The algorithms are comparable in total resource costs, as reflected by the fact that these two sample problems require similar degrees of error correction (hence very similar KQ product). The simulation algorithm is more compact than Shor’s, requiring, in particular, fewer logical qubits for distillation, which reflects the fact that this algorithm performs fewer arithmetic operations in parallel. However, Shor’s algorithm has a shorter execution time in this analysis. Both algorithms can be accelerated through parallelism if the quantum computer has more logical qubits to work with [101,105].


Precise timing and sequencing of operations are crucial to making an architecture efficient. In the framework we present here, an upper layer in the architecture depends on processes in the layer beneath, so that logical gate time is dictated by QEC operations, and so forth. This system of dependence of operation times is depicted for QuDOS in Fig. 17. The horizontal axis is a logarithmic scale in the time to execute an operation at a particular layer, while the arrows indicate fundamental dependence of one operation on other operations in lower layers. Examining Fig. 17, we see that the time scales increase as one goes to higher layers because a higher layer must often issue multiple commands to layers below. Using QuDOS as an example, the Virtual layer must construct a

TABLE V. Summary of the computing resources in a layered architecture based on the QuDOS platform, for Shor’s algorithm factoring a 1024-bit number (same implementation as in Ref. [43]) and the ground-state simulation of the molecule alanine (C3H7NO2) using firstquantized representation. Further details about the algorithms are provided in Appendix B.
Computing resource Shor’s algorithm (1024-bit) Molecular simulation (alanine)
Layer 5 Application qubits 6144 6650
Circuit depth (Toffoli) 1:68 � 108 1:27 � 109
Layer 4 Logical distillation qubits 66 564 15 860
Logical clock cycles 5:21 � 109 3:94 � 1010
Layer 3 Code distance 31 31
Error per lattice cycle 2:58 � 10�20 2:58 � 10�20
Layer 2 Virtual qubits 4:54 � 108 1:40 � 108
Error per virtual gate 1:00 � 10�3 1:00 � 10�3
Layer 1 Quantum dots 4:54 � 108 1:40 � 108
Area on chip 4:54 cm2 1:40 cm2
Estimated execution time 1:81 days 13:7 days

page 18

FIG. 17. Relative time scales for critical operations in QuDOS within each layer. Each bar indicates the approximate time scale of an operation, and the width indicates that some operation times may vary with improvements in technology. The gray arrows indicate dependence of higher operations on lower layers. The red arrow signifies that the surface-code lattice refresh must be 2–3 orders of magnitude faster than the dephasing time in order for error correction to function. The Application layer is represented here with a Toffoli gate, which is a common building block of quantum algorithms. Complete algorithm runtimes can vary significantly, depending on both the capabilities of the quantum computer and the specific way each algorithm is compiled, such as to what extent calculations are performed in parallel. The optical-control hardware is discussed in Appendix A.

virtual 1-qubit gate from a sequence of spin-state rotations. This process includes the duration of the laser pulses and the delays between pulses, which all add together for the total duration of the virtual gate. A crucial point shown in Fig. 17 is that the time to implement a logical quantum gate can be orders of magnitude greater than the duration of each individual physical process, such as a laser pulse. This increase in operation time is an important consideration for quantum-computer designs that rely on comparatively slower physical processes. At the same time, a quantum computer with a subset of very fast control mechanisms is limited by the slowest essential gate process, as QuDOS can only operate as fast as the 2-qubit entangling gate in Layer 1 permits. For large-scale quantum computing, the speed of logical, error-corrected operations is the crucial figure of merit. Figure 17 also highlights the fact that different control operations in the computer occur on substantially different time scales; achieving synchronization of these processes is an important function for a quantum-computer architecture. To facilitate this process, each layer in the architecture has an internal ‘‘clock frequency,’’ which is characteristic of the time scale of operations in that layer. These clock cycle times for each layer in QuDOS are listed in Table VI, along with the operations that define them. Even within the same layer, some processes may take different lengths of time to execute, so setting a clock cycle synchronizes these operations. Accordingly, as one layer builds on operations in a lower layer, the two layers are naturally synchronized. Synchronization alone is not sufficient for a quantum computer to function. Consider again the control cycle depicted in Fig. 2. Extracting and processing the error syndrome must be executed on time scales of the same

TABLE VI. Clock cycle times for Layers 1 to 4 in our analysis of QuDOS. The cycle time in each layer is determined by a fundamental control operation. Many operations possess some flexibility that would permit tradeoffs in execution time and system size, and better methods may be discovered.
Layer Clock cycle Limiting operation
4: Logical 30 �s Logical CNOT
3: QEC 256 ns Lattice refresh (syndrome circuit)
2: Virtual 32 ns Virtual 1-qubit gate
1: Physical 8 ns Laser-pulse repetition frequency

page 19 order as the duration of a logical gate or errors will accumulate faster than they can be detected. This function is performed by classical circuitry, but the required computing effort may not be trivial. Fast quantum operations can be a burden when error correction requires complex (classical) calculations, as is the case for the surface code. Devitt et al. [37] and Fowler et al. [96,123] examined this problem, finding that the processing requirements for surface-code error correction are not trivial; performing these calculations ‘‘live’’ where the results may be needed within e.g., 10 �s could be one of the more important problems for engineering a quantum computer. Still, the recent progress in this area suggests that some combination of improved algorithm software and custom hardware can achieve the necessary performance [123].


We have presented a layered framework for a quantumcomputer architecture. The layered framework has two major strengths: it is modular, and it facilitates fault tolerance. The layered nature of the architecture hints at modularity, but the defining characteristic of the layers we have chosen is encapsulation. Each of the layers has a unique and important purpose and bundles the related operations to fulfill this purpose. Additionally, each layer plays the role of resource manager, since often many operations in a lower layer are combined in a higher layer. Since technologies in quantum computing will evolve over time, layers may need replacement in the future, and encapsulation makes integration of new processes a more straightforward task. Fault tolerance is at present the biggest challenge for quantum computers, and our organization of layers is deliberately chosen to serve this need. Arguably, Layers 1 and 5 define any quantum computer, but the layers in between are devoted exclusively to creating fault tolerance in an intelligent fashion. Layer 2 uses simple control to mitigate systematic errors, so this layer is positioned close to the Physical layer where techniques like dynamical decoupling and decoherence-free subspaces are most effective. Layer 3 hosts quantum error correction (QEC), which is essential for large-scale circuit-model quantum computing on any hardware, such as executing Shor’s algorithm on a 1024-bit number. There is a significant interplay between Layers 2 and 3, because Layer 2 enhances the effectiveness of Layer 3. Finally, Layer 4 fills the gaps in the gate set provided by Layer 3 to form any desired unitary operation to arbitrary accuracy, thereby providing a complete substrate for universal quantum computation in Layer 5. QuDOS, a specific hardware platform we introduce here, demonstrates the power of the layered-architecture concept, but it also highlights a promising set of technologies for quantum computing, which are particularly noteworthy for the fast time scales of quantum operations, the high degree of integration possible with solid-state fabrication, and the adoption of several mature technologies from other fields of engineering. The execution times for fundamental quantum operations are discussed in Sec. IIG, but the importance of these fast processes becomes clear in Fig. 17, where the overhead resulting from virtual gates in Layer 2, QEC in Layer 3, and gate constructions in Layer 4 increases the time to implement quantum gates from nanoseconds in the Physical layer to milliseconds in the Application layer, or 6 orders of magnitude. In this context, a quantum computer needs very fast physical operations. One of our principal objectives is to better understand the resources required to construct a quantum computer that can solve a problem intractable for classical computers. Common figures of merit for evaluating quantumcomputing technology are gate fidelity, operation time, and qubit coherence time. This investigation goes further to show how connectivity and classical control performance are also crucial. Designing a quantum computer requires viewing the system as a whole, such that tradeoffs and compatibility between component choices must be addressed. A holistic picture is equally important for comparing different quantum-computing technologies, such as ion traps or superconducting circuits. This work illustrates how to approach the complete challenge of designing a quantum computer, so that one can adapt these techniques to develop architectures for other quantum-computing technologies we have not considered here. Using this design, differing system proposals can be compared within a common framework, which gives aspiring quantum engineers a common language for determining the best quantumcomputing technology for a desired application.


This work was supported by National Science Foundation Contract No. CCF-0829694, the University of Tokyo Special Coordination Funds for Promoting Science and Technology, NICT, and the Japan Society for the Promotion of Science (JSPS) through its Funding Program for World-Leading Innovative R&D on Science and Technology (FIRST Program). N. C. J. was supported by the National Science Foundation. A. G. F. acknowledges support from the Australian Research Council, the Australian Government, and from the U.S. National Security Agency (NSA) and the Army Research Office (ARO) under Contract No. W911NF-08-1-0527.


The QuDOS design depends on applying millions (or billions) of laser control pulses in parallel. For completeness, we outline here a method for achieving this level of control, but a detailed analysis of the engineering problem lies outside the scope of this work. Imagine that the 2D page 20

FIG. 18. Optical setup in QuDOS which controls many quantum dots in parallel. Phase-shift masks are used to produce diffractionlimited optical patterns which entangle the physical qubits (bottom). Computation is achieved by measuring the resulting cluster state; the measurement pattern is controlled by MEMS mirrors (top). Two MEMS-mirror arrays are multiplexed with electro-optic modulators, so that one can reposition mirrors while the other is ‘‘active.’’

array of quantum dots in a cavity is an image plane, like a projector screen. The challenge is to create a precisely controlled optical pattern on this screen. We need two key elements for this scheme to work: the ability to modulate laser pulses to each point on the screen (i.e., quantum dot) and the ability to focus laser signals near the diffraction limit. Similar concepts are presented in Ref. [5] To solve the first problem, we propose to use an array of MEMS mirrors. This technology was developed for high-definition projectors and optical switches for telecommunications [124,125], but the same devices are being adapted for use in quantum information processing [126,127]. Since MEMS mirrors are based on the same fabrication techniques as used for integrated circuits, a controllable mirror array with millions of units has been demonstrated commercially [124], and even larger arrays may be possible. The 2-qubit gate mechanism in Sec. IID requires the quantum dots to be in relatively close proximity (1 �m), which is close to the wavelength of the laser light (920 nm). Therefore, any optical patterns will have to compete with diffraction. This is a familiar problem in photolithography, so we propose to adopt the method of phase-shift masking [128,129] from that field. In essence, a ‘‘mask’’ is defined by a transparent plate patterned in such a way that light passing through the mask receives a phase shift that is a function of position within the image plane. This technique creates interference of light coming from different directions in such a manner that one can produce diffractionlimited patterns on the quantum-dot array. The MEMS mirrors and phase-shift masks work together as follows. Operations on the surface code follow a highly regular pattern of virtual gates; more specifically, the surface code can be constructed by building a cluster state and performing measurement on selected virtual qubits to create defects in the lattice [90]. The cluster-state operations are decomposed into a sequence of laser pulses, which are in turn created by appropriately designed phaseshift masks. Separately, a pattern of measurement pulses for each virtual qubit is modulated by a MEMS array. All of these optical signals are multiplexed together and sent to the quantum-dot array, which is depicted in Fig. 18. The configuration of defects in the surface code changes more slowly than one cycle of the syndrome extraction circuit. Because the defect boundaries must all be separated by the code distance d, the pattern of defects can be rearranged every d=4 lattice steps. Therefore, the MEMS mirrors, which control where measurements are made, can be rearranged every 2 �s in QuDOS, which is compatible with current technology [130]. Still, one has to account for the time required to reposition a set of mirrors. Two sets of mirrors are used in an alternating sequence: One is being repositioned while the other is actively in use, as shown in the top of Fig. 18. Electro-optic modulators can quickly multiplex laser pulses between the two mirror arrays.


We provide here a brief summary of the computational complexity for Shor’s algorithm and quantum simulation in first-quantized form. This analysis produces the resource estimates in Sec. VI.

1. Shor’s algorithm[edit]

We adopt the same implementation of Shor’s algorithm given in Van Meter et al. [43]. In order to determine the performance of Shor’s algorithm at Layer 5, we must look page 21

TABLE VII. Generation rates and maximal consumption rates for a 105-qubit quantum computer running Shor’s factoring algorithm. When the speed-of-data consumption rate is higher than the distillation rate, Shor’s algorithm experiences delays.
Bits to factor Ancilla-factory cross-ection (logical qubits) Distillation rate (jAi per cycle) Max. consumption rate (jAi per cycle)
512 96 928 84.1 32.1
1024 93 856 81.5 57.8
2048 87 712 76.1 105.1
4096 75 424 65.5 192.7
8192 50 848 44.1 355.7
16 384 1696 1.5 660.6

at how efficiently Layer 4 prepares Toffoli gates. Let us suppose that the quantum computer has capacity for 105 logical qubits; in general, one can interchange logical capacity and algorithm-execution time. To factor an N-bit number, approximately 6N application qubits are used by the algorithm itself, with the remainder of the logical qubits used to produce the crucial jAi ancillas. Implementations with fewer application qubits are possible [113,115], but the performance of such circuits is dramatically slower, especially if one is restricted to a limited set of gates. As shown in Sec.VB, one round of jAi distillation requires a volume of computing resources with crosssection of 12 logical qubits and time of 6 CNOT cycles. We arrange the excess (105 � 6N) qubits in factories which distill ancillas as fast as possible. As before, we define a Logical-layer clock cycle as one CNOT gate. We express the rate at which the factory generates ancillas by mean number of ancillas produced per clock cycle. Two levels of distillation will require 16 distillation circuits (15 at the first level, 1 at the second level), which uses a circuit volume of Vdistill ¼ 16 � ð12 logical qubitsÞ � ð6 clock cyclesÞ. For a given cross-section area Afactory of the quantum computer devoted to distillation, the maximal rate of ancilla production is given by Afactory=Vdistill. Calculations of these values are given in Table VII. We need to determine whether the quantum computer can run as fast as the circuit depth in Layer 5, or whether the distillation of jAi states limits performance. Using a construction like Fig. 14, the depth of the Toffoli gate is 31 clock cycles, where each S gate requires 4 cycles as shown in Fig. 12, and the circuit requires 7 distilled jAi ancillas. The circuit uses the carry-lookahead adder construction in Ref. [112], which requires approximately 10N Toffoli gates in total with a circuit depth of ðabout 4log2NÞtToffoli, or about 124log2N cycles. Using these figures, the maximal consumption rate of ancillas can be calculated, as shown in Table VII. As the size of the number to be factored increases, a fixed-size quantum computer is at some point unable to generate enough ancillas to run the algorithm at

FIG. 19. Circuit representation for one iteration of the Hamiltonian propagator in first-quantized form. The QFT is performed on the wave function, transforming between position basis and momentum basis.

maximum speed; when this happens, execution time is limited by the distillation process. One can make a crude estimate from Table VII that an efficient quantum computer for Shor’s algorithm must devote 90% of its resources to distillation. By similar arguments, a minimal-size quantum computer that holds just the algorithm qubits and distills one jAi ancilla at a time will be very slow.

2. Quantum simulation[edit]

We utilize the method in Ref. [100] to perform simulation in first-quantized form. Each electron wave function is represented on a three-dimensional Cartesian grid with 12 bits of precision in each dimension, which requires a quantum register of 36 qubits per particle.We elect to use a different set of adders and multipliers than Ref. [100], opting instead for simple ripple-carry adders which suffice for 12-bit precision [111]. First, the potential energy operator is calculated in the position basis. We transform the wave function representation from position basis to momentum basis with the quantum Fourier transform (QFT), allowing efficient evaluation of the kinetic energy operator. The inverse QFT transforms our system back to position basis. The quantum circuit representation of the system propagator U is depicted in Fig. 19. The resource requirements for each of the kinetic (T ), potential (V), and QFT operators are summarized in Table VIII. The parameters in Table VIII were derived assuming parallel calculation of commuting operator terms; for example, the Coulomb interaction between particles � and can be calculated simultaneously as

and �,

because these terms in the Hamiltonian commute and the circuits are disjoint [105]. Moreover, we have used the preceding analysis here in Appendix B 2 to include in

TABLE VIII. Resource requirements for the operators in firstquantized molecular simulation with B particles and 12-bit spatial precision including ancilla distillation.
Operator Maximum memory size (logical qubits) Circuit depth (Layer 4 clock cycles)
Kinetic cnergy 334 � B 1:55 � 105
Potential cnergy 369 � B 6:26 � 105 � B
QFT 272 � B 2:57 � 104

page 22

FIG. 20. The time evolution of the Hamiltonian is produced by iterating the system propagator over many time steps. After evolution, a quantum Fourier transform of the time vector transforms the system into the energy eigenbasis, allowing readout of an energy eigenvalue.

these figures the size of ancilla factories, which is approximately 260B logical qubits in order to simulate a system of B particles. For this parallel-simulation algorithm, ancilla production consumes about 70% of the quantum computer. The circuit construction in Fig. 19 is just one iteration of the system propagator. Estimating an energy eigenvalue requires simulation of the system at discrete time steps, so the propagator is repeated many times [99], as shown in Fig. 20. After evolving the propagator along these time steps, the system is transformed to the energy eigenbasis by means of a QFT operation on the time vector jti [54]. The precision in the final answer is limited by the number of bits in jti, so for this analysis we assume the system is evolved for 210 time steps, which offers at most about 3 decimal digits of precision.

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Layered Architecture for Quantum Computing
N. Cody Jones, Rodney Van Meter, Austin G. Fowler, Peter L. McMahon, Jungsang Kim, Thaddeus D. Ladd, and Yoshihisa Yamamoto
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Phys. Rev. X 2, 031007 – Published 31 July 2012