Logical blocks for fault-tolerant topological quantum computation

Abstract

Logical gates constitute the building blocks of fault-tolerant quantum computation. While quantum errorcorrected memories have been extensively studied in the literature, explicit constructions and detailed analyses of thresholds and resource overheads of universal logical gate sets have so far been limited. In this paper, we present a comprehensive framework for universal fault-tolerant logic motivated by the combined need for (i) platformindependent logical gate definitions, (ii) flexible and scalable tools for numerical analysis, and (iii) exploration of novel schemes for universal logic that improve resource overheads. We first introduce the theory of faulttolerant logical channels for describing logical gates holistically in space-time. Focusing on channels based on surface codes, we introduce explicit, but platform-independent representations of topological logic gates-called logical blocks-and generate new, overhead-efficient methods for universal quantum computation. As a specific example, we propose fault-tolerant schemes based on surface codes concatenated with more general low-density parity check (LDPC) codes, suggesting an alternative path toward LDPC-based quantum computation. The logical blocks framework enables a convenient software-based mapping from an abstract description of the logical gate to a precise set of physical instructions for executing both circuit-based and fusion-based quantum computation (FBQC). Using this, we numerically simulate a surface-code-based universal gate set implemented with FBQC, and verify that the threshold for fault-tolerant gates is consistent with the bulk threshold for memory. We find, however, that boundaries, defects, and twists can significantly impact the logical error rate scaling, with periodic boundary conditions potentially halving the memory resource requirements. Motivated by the favorable logical error rate suppression for boundaryless computation, we introduce a novel computational scheme based on the teleportation of twists that may offer further resource reductions.

Classification of small triorthogonal codes

“…I. List of representative polynomials of affine equivalence classes of RM (3,6) with weight less than or equal to 18. This list has been constructed using a computer search over the polynomials of the form x 1 g(x 3 , x 4 , x 5 , x 6 ) + x 2 h(x 3 , x 4 , x 5 , x 6 ) + x 1 x 2 u(x 3 , x 4 , x 5 , x 6 ), with deg g, deg h 2, and deg u 1.…”

“…Clifford operations can be implemented fault-tolerantly using Pauli stabilizer codes, and magic states of high fidelity can be distilled using Clifford operations. This way of achieving fault-tolerant quantum universality underlies leading proposals for quantum computers at scale [3][4][5][6]. However, the fault tolerance for non-Clifford operations via magic state distillation is estimated to be more costly than that for Clifford operations, and hence there have been many proposals to reduce the cost; see [4,7] and references therein.…”

Qudit lattice surgery

“…We have no new results or proofs in this section, but we would like to discuss the diagrams of logical blocks. These sorts of diagrams for lattice surgery have been used in an engineering setting to compile quantum circuits to lattice surgery [11,25]. To go from the cubes shown there to the tubes which we show here we merely relax the discretisation of space and time somewhat to expose the relationship with algebra.…”

Differentiating and Integrating ZX Diagrams

“…Moreover, it is possible to directly evaluate the tensor by performing local rewrites. Using these local rewrites, ZX-calculus has been successfully applied to circuit compilation [3,4,16,30], measurement-based quantum computing [22,17], fusion-based quantum computing [6], quantum error correction [5], quantum natural language processing [12,25,21], and quantum foundations [1,11,10,18]. ZX-calculus can even be used as a concrete realisation of quantum theory [13].…”