Fault-Tolerant Benchmark Applications

This page collects benchmark circuits and experiments for fault-tolerant quantum computing. Each section focuses on a different application family and lists concrete implementations with code parameters, noise / decoder assumptions, and basic performance metrics.

The tables below use the following columns:

  1. Name – short label for the benchmark instance
  2. Code (type, parameters) – e.g., surface code d=3, Steane [[7,1,3]]
  3. Noise / Decoder – noise model and decoding strategy
  4. Qubits – total physical qubit count (data + ancilla/helper)
  5. Fidelity – logical success probability or 1 − logical error rate
  6. GitHub – link to code or configuration

1. Logical Gates

Logical gate benchmarks study encoded single- and two-qubit gates (e.g., logical X, Z, H, S, T, CNOT) implemented on top of a quantum error-correcting code. These benchmarks are useful for comparing different codes, layouts, and decoders under the same physical noise assumptions.

Benchmark Instances

Name Code (type, parameters) Noise / Decoder Qubits Fidelity GitHub
Logical CNOT (Surface d=3) Surface code, rotated lattice, distance d=3 Depolarizing p=1e-3 (2-qubit), MWPM decoder (PyMatching) 49 0.992 Link
Logical T via MSD (Steane) Steane code [[7,1,3]] with magic-state distillation Biased dephasing p_Z = 2e-3, lookup-table decoder 35 0.985 Link
Repeated Logical Id (Bacon–Shor) Bacon–Shor code n_x=3, n_z=3 Circuit-level depolarizing p=5e-4, belief-propagation decoder 27 0.998 Link

Replace USERNAME/REPO* with your actual repositories or configs.

2. Shor’s Algorithm

Shor’s algorithm factors an integer using modular exponentiation and a quantum Fourier transform (QFT). Fault-tolerant benchmarks typically use small composite numbers (e.g., 15, 21) encoded in QEC codes and focus on the cost and reliability of modular arithmetic plus QFT under realistic noise.

Benchmark Instances

Name Code (type, parameters) Noise / Decoder Qubits Fidelity GitHub
Shor N = 15 (Logical) Surface code d=3 for all logical qubits Circuit-level depolarizing p=1e-3, MWPM decoder 128 0.93 Link
Shor N = 21 (Hybrid Layout) Mixed surface codes d=3 / d=5 for work registers Spatially correlated noise (p_2q=1e-3), tensor-network decoder 256 0.89 Link
Shor N = 15 (Color Code) 2D color code distance d=3 Biased dephasing (p_Z > p_X), MWPM-style color decoder 162 0.91 Link

3. QAOA

The Quantum Approximate Optimization Algorithm (QAOA) is a variational algorithm for combinatorial optimization (e.g., MaxCut, Max-2-SAT). Fault-tolerant QAOA benchmarks emphasize the overhead of implementing parameterized layers at logical level, including logical ZZ and X mixers, and measure how logical error rates affect the final approximation ratio.

Benchmark Instances

Name Code (type, parameters) Noise / Decoder Qubits Fidelity GitHub
QAOA MaxCut (p = 1, 8 nodes) Surface code d=3 (one logical per graph vertex) Depolarizing p=1e-3, MWPM decoder 96 0.95 Link
QAOA MaxCut (p = 2, 12 nodes) Surface code d=5, lattice-surgery entangling gates Circuit-level depolarizing p=5e-4, MWPM decoder 320 0.92 Link
QAOA Portfolio Opt. (p = 1) Steane [[7,1,3]] logical qubits for cost Hamiltonian Asymmetric noise (T_1, T_2-derived channels), lookup-table decoder 70 0.90 Link

4. Grover’s Algorithm

Grover’s search algorithm provides quadratic speedup for unstructured search. Fault-tolerant Grover benchmarks typically implement an oracle and diffusion operator using logical gates, then measure how many Grover iterations can be performed before logical errors dominate.

Benchmark Instances

Name Code (type, parameters) Noise / Decoder Qubits Fidelity GitHub
Grover 1-marked item (n = 4) Surface code d=3, logical oracle via Toffoli synthesis Depolarizing p=1e-3, MWPM decoder 80 0.96 Link
Grover 1-marked item (n = 6) Surface code d=5, helper-qubit-heavy ancilla factory Correlated 2-qubit noise, tensor-network decoder 240 0.93 Link
Grover oracle with QEC-protected data Bacon–Shor logical data, unencoded ancilla helpers Amplitude-damping + dephasing, belief-propagation decoder 112 0.90 Link

You can:

  • Adjust intros to match your own terminology.
  • Replace the placeholder repos (USERNAME/REPO*) with your actual GitHub links.
  • Add or remove rows in each table as your benchmark suite grows.