Efficient Classical Computation of Single-Qubit Marginal Measurement Probabilities to Simulate Certain Classes of Quantum Algorithms

TL;DR

The paper improves classical QC-DFT simulations of quantum circuits by introducing a neural-network-assisted CNOT functional that uses an intermediary unitary gate $\hat{U}_m = e^{iH(\boldsymbol{\theta})}$, where $H(\boldsymbol{\theta}) = \sum_{\sigma_i,\sigma_j \in P} \theta_{ij} \sigma_i \otimes \sigma_j$ with $P = \{I, \sigma_x, \sigma_y, \sigma_z\}$. The parameters \boldsymbol{\theta} are predicted from 1-RDMs via neural networks and trained with an RMS1F infidelity loss to minimize the distance between predicted and exact 1-RDMs, enabling more accurate single-qubit marginal probabilities (SQPs) while preserving constant-time, linear-space scaling in circuit size. Results show lower SQP error and higher mean fidelity for the proposed method compared to Bernardi's QC-DFT on random circuits, with similar runtimes, though fundamental limitations remain in capturing entanglement and joint probabilities. The work highlights both practical improvements for certain problem classes (e.g., specific Grover/Shor scenarios) and the broader challenge of representing entanglement in 1-RDM-based simulations, providing a pathway for targeted, tractable classical approximations of quantum circuits.

Abstract

Classical simulations of quantum circuits are essential for verifying and benchmarking quantum algorithms, particularly for large circuits, where computational demands increase exponentially with the number of qubits. Among available methods, the classical simulation of quantum circuits inspired by density functional theory -- the so-called QC-DFT method, shows promise for large circuit simulations as it approximates the quantum circuits using single-qubit reduced density matrices to model multi-qubit systems. However, the QC-DFT method performs very poorly when dealing with multi-qubit gates. In this work, we introduce a novel CNOT "functional" that leverages neural networks to generate unitary transformations, effectively mitigating the simulation errors observed in the original QC-DFT method. For random circuit simulations, our modified QC-DFT enables efficient computation of single-qubit marginal measurement probabilities, or single-qubit probability (SQPs), and achieves lower SQP errors and higher fidelities than the original QC-DFT method. Despite some limitations in capturing full entanglement and joint probability distributions, we find potential applications of SQPs in simulating Shor's and Grover's algorithms for specific solution classes. These findings advance the capabilities of classical simulations for some quantum problems and provide insights into managing entanglement and gate errors in practical quantum computing.

Figures (4)

• Figure 1: Evolution of (a) SQP error, (b) mean fidelity, and (c) mean SQP calculated with our proposed CNOT functional (solid line) compared with the original QC-DFT method by Bernardi (dashed line). In panels (a) and (b) the dash-dotted lines show the differences between the two methods, while gray lines are a guide for eyes to show zeros. In panel (c), the horizontal solid line indicates the value of converged mean SQP at 0.5 (or 50%).
• Figure 2: The success rate of simulating Shor's algorithm using SQPs. (a) The number of integers $1 \leq a_s \leq n_S$ for which the period of $a_s \mod N_s$ satisfies $r = 2^y$ and $\gcd\left(a_s^{r/2} + 1, N_s\right)$ yields a non-trivial factor for some $N_s$. (b) The probability of sampling a value $a_s$ from arbitrary integers $a \in \left[1, N_s\right]$ for the first 3000 squarefree semiprimes $N_s$.
• Figure S1: The evolution of (a) the loss function and (b) the mean fidelity during the training of the neural networks.
• Figure S2: Time Benchmark for the three different methods. Each data point is the average time it takes to apply 10 random gates for 10 circuits.