N. Cody Jones, Rodney Van Meter, Austin G. Fowler, Peter L. McMahon, Jungsang Kim, Thaddeus D. Ladd, and Yoshihisa Yamamoto

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ B. Virtual gate 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. ”

“ 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. ”

“ B. Layered framework 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. ”

“ A. Prior work on quantum-computer architecture Many different quantum-computing technologies are under experimental investigation [1] , but for each a scalable system architecture remains an open research 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] . ”

“ 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. ”

“ 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. ”

“ 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] . ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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 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] . ”

“ 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. ”

“ 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. ”

“ 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] . ”

“ 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] . ”

“ 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. ”

“ 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. ”

“ 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] . ”

“ 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] . ”

“ 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. ”