Benchmarking Techniques for Decoded Quantum Interferometry

Abstract

We develop a new benchmarking scheme for the Decoded Quantum Interferometry (DQI) algorithm quantifying the number of quantum gates required to obtain an optimal solution to a problem amenable to DQI. We apply the benchmarking scheme to the Binary Paint Shop Problem (BPSP) in order to benchmark the performance of DQI against a state of the art classical solver. To do so, we provide an explicit construction of a quantum circuit implementation of a greedy decoder for low-density parity check codes arising from max-2-XORSAT problems.

Figures (8)

• Figure 1: Testing the approximation scheme for the benchmarking procedure with the BPSP. For each number of car pairings $\{5,7,\dots, 15\}$, we randomly generate $10$ BPSP instances and compute $p_{Opt}$ and $\tilde{p}_{Opt}$. The BPSP and the greedy decoder are explained in detail in Section \ref{['sec_BPSP']}. The lighter areas correspond to the standard deviation of the obtained results.
• Figure 2: Code distance for a number of randomly generated BPSP instances. For both $100$ and $1000$ car pairings, we randomly generate $100$ instances each and compute the corresponding code distances $d^\bot$. We note that the code distance $d^\bot$ does not increase meaningfully with the higher number of car pairings.
• Figure 3: Comparison of failure rates for a minimum-length decoder and our implementation of the greedy decoder. For each number of car pairings, we generate $10$ random BPSP instances and decode $2000$ randomly generated errors per Hamming weight and instance.
• Figure 4: Quantum circuit for greedy decoding of the BPSP instance \ref{['BPSP_example']}. The decoding part of the circuit comprises one block for each path in $P$. Subsequently, the information stored in the auxiliary register is used to restore the syndrome register.
• Figure 5: Leading order contribution to the number of quantum gates needed for the quantum circuit of the greedy decoder described in algorithm \ref{['quantum_circuit_generation']}. We randomly generate $10$ BPSP instances for each number of car pairings. The two other curves are added to make it easier to visualize the growth rate of this leading order contribution. The light blue area around the blue graph (barely visible) corresponds to the standard deviation of the distribution of the obtained results.

Theorems & Definitions (1)

• Definition 3.1