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Circuit Design and Efficient Simulation of Quantum Inner Product and Empirical Studies of Its Effect on Near-Term Hybrid Quantum-Classic Machine Learning
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Designs quantum inner product (QIP) circuits for 1-to-1, 1-to-N and M-to-N cases, and a classical state simulator (torch_qip) that directly simulates their output states, enabling evaluation of QIP in ML tasks.
Problem
Quantum inner product circuits had only theoretical treatment; no explicit, implementable circuits or practical evaluation in hybrid quantum-classical ML under NISQ constraints existed, and simulating such circuits classically was intractable.
Approach
Builds explicit QIP circuits by combining the Hadamard test with quantum phase estimation for vector, linear-mapping and matrix-multiplication cases, and simulates their output state distribution directly (with quantum state search and result reuse) instead of simulating gates, yielding differentiable QIP operators for PyTorch.
Key results
- Simulator speeds up the 1-to-1 QIP circuit simulation by 68251x at d=64 (t=6) versus qiskit while giving numerically the same results.
- On MNIST, Q-8-Mode reaches 96.13% test accuracy versus 95.97% for the classical model after 20 epochs.
- K-Means with QIP underperforms the classical model even with 8 qubits (e.g., NMI 0.7300 vs 0.7314 for Q-8-Mode is close, but ranking-sensitive QIP degrades in general per the authors' analysis), and IBM Brisbane errors are much higher than the simulator's.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use QIP with 8 qubits and OUTPUT-MODE in ProjUNN-D | ProjUNN-D on MNIST · MNIST test | accuracy (%): 95.97 -> 96.13 | Helped |
| Use QIP in K-Means with 4 qubits and OUTPUT-MODE | K-Means · MNIST | RI: 0.9318 ± 0.0096 -> 0.7866 ± 0.0079 | Hurt |
| Use QIP in K-Means with 8 qubits and OUTPUT-MODE | K-Means · MNIST | NMI: 0.7314 ± 0.0156 -> 0.7300 ± 0.0229 | No clear effect |
| Use QIP in K-Means with 8 qubits and OUTPUT-MODE | K-Means · MNIST | AMI: 0.7211 ± 0.0191 -> 0.7185 ± 0.0268 | No clear effect |
| Run QIP on IBM Brisbane rather than the circuit simulator | 1-to-1 QIP · d=2 | MSE: 0.0066 ± 0.0098 -> 0.0990 ± 0.1835 | Hurt |
| Replace circuit simulation with direct output-state simulation | QIP simulator · d=64 | running time: 2388.785 -> 0.035, 68251x faster | Helped |
| Add quantum state search (ablation) | QIP matrix-multiplication simulation · t=6, r=1 | running time: 44.86 s -> 4.74 s (9.45x) | Helped |
| Add quantum state search (ablation) | QIP matrix-multiplication simulation · t=2, r=1 | running time: 3.44 s -> 3.46 s (0.99x) | No clear effect |
| Increase the number of QIP output qubits (ablation) | QIP · Random inner products | MSE: 0.2665±0.0027 -> 0.0038±0.0002 | Helped |
| Increase QIP sampling times (ablation) | QIP using OUTPUT-AVG · Random inner products, t=2 | MSE: 0.2665±0.0027 -> 0.1034±0.0005 | Helped |
| Use OUTPUT-MODE instead of OUTPUT-AVG at t=2, r=7 (ablation) | QIP · Random inner products | MSE: 0.1034±0.0005 -> 0.1462±0.0004 | Mixed |
| Use OUTPUT-MODE instead of OUTPUT-AVG at t=8, r=7 (ablation) | QIP · Random inner products | MSE: 0.0006±0.0000 -> 0.0000±0.0000 | Helped |
Surprising findings
- Q-8-Mode quantum training slightly outperforms the classical implementation on MNIST, which the authors attribute to quantum-induced randomness acting like dropout.
- IBM Brisbane hardware showed substantially higher MSE/MAE than the classical circuit simulator, indicating current hardware is not yet ready for QIP.
Limitations
- QIP only accepts normalized vectors as input.
- Noise and decoherence are non-negligible for deep circuits on real hardware.
- K-Means clustering is sensitive to QIP errors and degrades relative to classical.
Experimental setup
Datasets: MNIST PPI BlogCatalog uniform random vectors (100K inner product instances)
Benchmarks: MNIST PPI BlogCatalog
Metrics: MSE MAE test accuracy Macro-F1 Micro-F1 RI NMI AMI running time
Baselines: Qiskit circuit simulator classical inner product ProjUNN-T classical K-Means
Builds on: Hadamard test quantum phase estimation ProjUNN node2vec
Closest papers by meaning
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