Quantum Protocol Gets Scalability with Hybrid QLDPC Codes
A New Quantum Computing Protocol Reduces Overheads
Fault-tolerant quantum computation (FTQC) has struggled to reduce resource overheads needed to build practical quantum computers. Shiro Tamiya, Masato Koashi, and Hayata Yamasaki discovered a fault-tolerant protocol with constant space overhead and polylogarithmic time overhead. Concatenated Steane codes and non-vanishing-rate quantum low-density parity-check (QLDPC) codes are used in this novel protocol.
Challenge of Fundamental Overheads
FTQC is essential because quantum systems are brittle and require quantum error-correcting codes to encode logical qubits. Space overhead—the amount of physical qubits needed per logical qubit—and time overhead—the ratio of physical circuit depth to logical circuit depth—are the main costs of this approach.
Conventional FTQC systems, such as those using surface codes or concatenated Steane codes, have polylogarithmic space and time overheads. Recent years have seen advances in non-vanishing-rate codes for constant space overhead. This innovation incurred greater temporal overheads that sometimes scaled super-polylogarithmically. Previous protocols that used QLDPC codes for constant space overhead required sequential gate implementation, which added polynomial time overhead.
Newer, truly parallel approaches had just quasi-polylogarithmic time overhead. This study sought to overcome super-polylogarithmic time overhead while maintaining constant space overhead.
Hybrid QLDPC/Concatenated Code Protocol
A mixed protocol is advised. Non-vanishing-rate QLDPC codes, specifically quantum expander codes, protect logical qubits as quantum memory. Quantum expander codes are suited for resilient quantum memory because their settings reduce decoding failure exponentially with code distance and maintain a positive code rate (logical to physical qubits).
The protocol uses concatenated Steane codes to perform global QLDPC memory operations. The QLDPC code's logical auxiliary states are prepared fault-tolerantly using this well-established method. After these QLDPC-encoded auxiliary states are ready, fault-tolerant gate teleportation can build logical Clifford and non-Clifford gates.
More gate parallelism than current QLDPC-based protocols is essential for polylogarithmic time overhead. A more detailed examination shows that the required QLDPC code block size can be quite tiny, scaling just polylogarithmically. By reducing auxiliary state preparation resources, this tiny scale promotes parallelism and achieves polylogarithmic time overhead while keeping constant space overhead. This lets many logical operations run at each time step.
Complete the Threshold Proof: Partial Circuit Reduction
QLDPC protocols struggled with the threshold theorem proof's rigour. Previous studies often neglected error correlations with surrounding circuitry to focus on local error suppression within a QLDPC code block. This error left the threshold theory proof logically incomplete.
This was addressed by the researchers' unique, meticulous partial circuit reduction strategy. This method enables modular fault-tolerant circuit error analysis. The proof is done in steps by looking at small portions of the circuit, called “rectangles,” which are made up of operation and error correction devices.
Partial circuit reduction allows researchers to update the probability distribution of problematic sites for future circuit components while replacing a noisy rectangle with a noiseless, perfectly functional one. This systematic approach proves the threshold theorem for constant-space-overhead protocols with QLDPC codes by tightly regulating error correlations and using well-known decoding algorithms as black-box components.
Setting a Real-World Threshold
Another major theoretical advance was the explicit inclusion of non-zero classical computing time. Many earlier computations assumed immediate classical processing for gate teleportation and error correction (decoding). This operation has a non-zero runtime that could generate mistakes and slowdowns that make FTQC unfeasible if left unchecked.
By considering classical processing time, the analysis clearly compensates for this non-zero runtime. The QLDPC code decoding method must meet certain parameters to evaluate fault-tolerance in this practical scenario.
The small-set-flip decoder used for quantum expander codes fulfils these requirements, including the ability to fix errors in one shot and maintain a consistent execution time even when using parallel processors for classical processing. By thoroughly establishing the threshold under this realistic limitation, overheads may be understood for every conceivable bottleneck.
The QLDPC-code-based method achieves FTQC with a negligible slowdown and a bounded overhead of physical qubits, providing a solid foundation for future high-performance quantum computation.
