A Game of Surface Codes: Large-Scale Quantum Computing with Lattice Surgery

Abstract

Given a quantum gate circuit, how does one execute it in a fault-tolerant architecture with as little overhead as possible?In this paper, we discuss strategies for surface-code quantum computing on small, intermediate and large scales. They are strategies for space-time tradeoffs, going from slow computations using few qubits to fast computations using many qubits. Our schemes are based on surface-code patches, which not only feature a low space cost compared to other surface-code schemes, but are also conceptually simple -simple enough that they can be described as a tile-based game with a small set of rules. Therefore, no knowledge of quantum error correction is necessary to understand the schemes in this paper, but only the concepts of qubits and measurements.The field of quantum computing is fuelled by the promise of fast solutions to classically intractable problems, such as simulating large quantum systems or factoring large numbers. Already ∼100 qubits can be used to solve useful problems that are out of reach for classical computers [1,2]. Despite the exponential speedup, the actual time required to solve these problems is orders of magnitude above the coherence times of any physical qubit. In order to store and manipulate quantum information on large time scales, it is necessary to actively correct errors by combining many physical qubits into logical qubits using a quantum errorcorrecting code [3][4][5]. Of particular interest are codes that are compatible with the locality constraints of realistic devices such as superconducting qubits, which are limited to operations that are local in two dimensions. The most prominent such code is the surface code [6,7].Working with logical qubits introduces additional overhead to the computation. Not only is the space cost drastically increased as physical qubits are replaced by logical qubits, but also the time cost increases due to the restricted set of accessible logical operations. Surface codes, in particular, are limited to a set of 2Dlocal operations, which means that arbitrary gates in a quantum circuit may require several time steps instead of just one. To keep the cost of surface-code quantum computing low, it is important to find schemes that translate quantum circuits into surface-code layouts with a low space-time overhead. This is also necessary to benchmark how well quantum algorithms per-form in a surface-code architecture.There exist several encoding schemes for surface codes, among others, defect-based [7], twist-based [8] and patch-based [9] encodings. In this work, we focus on the latter. Surface-code patches have a low space overhead compared to other schemes, and offer lowoverhead Clifford gates [10,11]. In addition, they are conceptually less difficult to understand, as they do not directly involve braiding of topological defects. Designing computational schemes with surface-code patches only requires the concepts of qubits and measurements. To this end, we describe the operations of surface-code patches as a tile-based game. This is helpful t...

Qubitization of Arbitrary Basis Quantum Chemistry Leveraging Sparsity and Low Rank Factorization

“…As a consequence, the first quantum algorithms in this representation had gate complexity O(N 11 ) [12, 13]. Since then, a large community of researchers has worked to significantly reduce the cost of simulation in this representation through tighter bounds [13][14][15], better mappings between fermions and qubits [16][17][18][19][20], improved state preparation techniques [21][22][23][24], application of new time-evolution strategies [25][26][27], considerations of fault-tolerant overheads [28][29][30] and other representational and algorithmic insights [31][32][33][34][35][36].The lowest rigorous complexity of prior work on second quantized arbitrary basis chemistry simulation is either the O(N 5 ) scaling of [26], or the O(λ 2 ) scaling of [27], where λ is the 1-norm of the Hamiltonian. However, the [26] algorithm suffers from large constant factors in the scaling, and the approach of [27] scales quadratically worse than post-Trotter methods with respect to the evolution time.…”

“…Here we provide an algorithm with O(N 3/2 λ) T complexity, which appears better than all prior work so long as λ = Ω(N 3/2 ), which is usually the case.Prior papers to compile a quantum chemistry algorithm to the level of Clifford + T gates and estimate the resources required within an error-correcting code are [8, 9,29]. These papers focus on minimizing T complexity or Toffoli complexity because these gates cannot be transversely implemented within practical codes [30,37]. To implement these gates one must distill magic states or Toffoli states, which takes orders of magnitude more spacetime volume (qubitseconds) than executing Clifford gates and also consumes a very large number of physical qubits [38,39].The work of [29] focused on the simulation of an active space of the FeMo cofactor of the Nitrogenase enzyme, aka "FeMoco" (stoichiometry Fe 7 MoS 9 C).…”

Magic State Distillation: Not as Costly as You Think

“…Using the construction from Ref. [26], the entire computation can be finished in n T · d full code cycles, if 231 distance-d full surface-code patches are used to store the 100 qubits. The probability that any of these qubits is affected by a storage error that can spoil the outcome of the computation is p storage error = 231 · n T · d full · p L (p phys , d full ) .…”

Clifford recompilation for faster classical simulation of quantum circuits

“…Set ω ← (bij/|bij|)ω; We remark that the use of Pauli exponential version of Clifford and non-Clifford gates has recently found application in the context of error-correction [15].…”