[论文解读] Computing Efficiently in QLDPC Codes

Alexander J. Malcolm, Andrew N. Glaudell+11 more|ArXiv.org|Feb 11, 2025

Numerical Methods and Algorithms被引用 3

一句话总结

论文介绍 SHYPS（Subsystem HYpergraph Product Simplex）QLDPC 码，能够实现深度高效、容错的 Clifford 操作与接近内存的电路性能，降低 universial 量子计算的量子比特开销。

ABSTRACT

It is the prevailing belief that quantum error correcting techniques will be required to build a utility-scale quantum computer able to perform computations that are out of reach of classical computers. The QECCs that have been most extensively studied and therefore highly optimized, surface codes, are extremely resource intensive in terms of the number of physical qubits needed. A promising alternative, QLDPC codes, has been proposed more recently. These codes are much less resource intensive, requiring significantly fewer physical qubits per logical qubit than practical surface code implementations. A successful application of QLDPC codes would therefore drastically reduce the timeline to reaching quantum computers that can run algorithms with proven exponential speedups like Shor's algorithm and QPE. However to date QLDPC codes have been predominantly studied in the context of quantum memories; there has been no known method for implementing arbitrary logical Clifford operators in a QLDPC code proven efficient in terms of circuit depth. In combination with known methods for implementing T gates, an efficient implementation of the Clifford group unlocks resource-efficient universal quantum computation. In this paper, we introduce a new family of QLDPC codes that enable efficient compilation of the full Clifford group via transversal operations. Our construction executes any m-qubit Clifford operation in at most O(m) syndrome extraction rounds, significantly surpassing state-of-the-art lattice surgery methods. We run circuit-level simulations of depth-126 logical circuits to show that logical operations in our QLDPC codes attains near-memory performance. These results demonstrate that QLDPC codes are a viable means to reduce the resources required to implement all logical quantum algorithms, thereby unlocking a reduced timeline to commercially valuable quantum computing.

研究动机与目标

• Motivate the use of quantum LDPC codes (QLDPC) as a lower-qubit-overhead alternative to surface codes for universal quantum computation.
• Propose a new SHYPS code family combining SHP with simplex codes to enable many transversal Clifford operations.
• Show that arbitrary m-qubit Clifford operations can be compiled in depth O(1) per block, with overall depth 4bk(1+o(1)) for b blocks.
• Demonstrate via circuit-level simulations that SHYPS codes achieve near-memory performance under circuit-level noise.
• Discuss implications for practical quantum computing timelines and resource savings compared to surface codes.

提出的方法

• Construct SHYPS codes by coupling subsystem hypergraph product (SHP) with classical simplex codes.
• Use automorphisms of the SHYPS code to implement CNOT and diagonal Clifford operators with depth-1 physical circuits.
• Decompose Clifford circuits into DZ-CX-DX-DZ(1) layers to bound depth for arbitrary Clifford operators.
• Analyze Clifford compiling ratio and show it is O(1) for SHYPS, i.e., depth scales linearly with qubits, not with code distance.
• Perform circuit-level simulations to compare memory performance against scaled surface codes and to test Clifford circuits up to depth 126.

实验结果

研究问题

• RQ1 Can SHYPS codes support efficient, fault-tolerant Clifford operations via transversal or low-depth constructions?
• RQ2 What is the asymptotic depth required to implement arbitrary Clifford operators on SHYPS codes compared to surface-code implementations?
• RQ3 How do the SHYPS automorphisms and ZX-dualities translate into practical, fault-tolerant logical operations?
• RQ4 Do SHYPS codes exhibit near-memory performance under circuit-level noise in simulations?
• RQ5 What are the resource (time/space) costs for representative Clifford operations on SHYPS codes?

主要发现

Logical Gate	Time cost	Space cost
CNOT (cross-block)	4	0
CNOT (in-block)	4	n
S (in-block)	6	0
CZ (cross/in-block)	4	0
H (in-block)	8	0
Arbitrary b-block Clifford	4bk(1+o(1))	bn or 0

• SHYPS codes enable arbitrary Clifford operations across b blocks of k logical qubits in depth 4bk(1+o(1)).
• A single code block SHYPS provides O(1) time for elementary Clifford gates with zero or minimal additional space overhead.
• For cross-block CNOT, the depth is 4 with zero space overhead, and in-block CNOT depth is 4 with space proportional to n.
• Table 1 shows concrete costs: CNOT (cross-block) 4 time, 0 space; CNOT (in-block) 4 time, n space; S (in-block) 6 time, 0 space; CZ (cross/in-block) 4 time, 0 space; H (in-block) 8 time, 0 space; Arbitrary b-block Clifford 4bk(1+o(1)) time, bn or 0 space