Computing Efficiently in QLDPC Codes

TL;DR

This work introduces SHYPS, a family of quantum LDPC codes designed from highly symmetric simplex-classical codes to enable efficient, fault-tolerant Clifford computation with transversal-like gates. By leveraging code automorphisms and symplectic representations, the authors construct depth-1 generators for CNOTs and diagonal Cliffords, and fold-transversal Hadamards, yielding an overall Clifford compilation depth that scales as 4bk(1+o(1)) for b blocks, independent of code distance. Circuit-level simulations demonstrate near-memory performance for SHYPS circuits on modest qubit counts, with significant reductions in space-time overhead relative to surface codes, and memory-like error suppression for depth-126 Clifford circuits. The results establish SHYPS as a viable path toward low-overhead universal quantum computation, offering strong fault-tolerance guarantees and practical scalability, while outlining avenues to improve rates and parallelism further via measurement strategies. Overall, the paper argues that QLDPC codes, when equipped with structured automorphisms and efficient Clifford compilation, can dramatically accelerate the timeline to commercially valuable quantum computing with substantial resource benefits over conventional surface codes.

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.

Figures (12)

• Figure 1: Gauge generator connectivity for $r=3$ SHYPS code. Black circles denote data qubits. Orange (blue) squares indicate auxiliary qubits used to measure $X$ ($Z$) gauge generators. Each gauge generator has weight 3 and is measured using three CNOT gates, depicted as edges connecting the corresponding data qubits to a single auxiliary qubit. Two representative gauge generators are illustrated: one $X$-type and one $Z$-type. The full set of gauge generators can be obtained by translating this pattern under periodic boundary conditions.
• Figure 2: Simulation results for quantum memories under circuit-level noise for SHYPS and surface codes. For these simulations, only $Z$-type detectors are used.
• Figure 3: Simulation results for a depth-$126$ logical quantum circuit and a memory composed from two code blocks of the $[49, 9, 4]$ SHYPS code. $Z$- and $X$-type detectors are used in both cases.
• Figure 4: Physical implementation of a cross-block CNOT operator utilising $\pi \in {\mathrm{Aut}}(\mathcal{C})$.
• Figure 5: Space-time volume $V$ of a logical $m$-qubit Clifford operator performed on SHYPS codes, rotated surface codes using transversal operators, and rotated surface codes using lattice surgery for distances 8, 16 and 32.

Theorems & Definitions (131)

• Example 4.1
• Example 4.2
• Lemma 5.1
• Theorem 5.2
• Example 8.1
• Example 8.2
• Example 8.3
• Definition 8.4
• Definition 8.5
• Lemma 8.6