Enhancing Chemistry on Quantum Computers with Fermionic Linear Optical Simulation

TL;DR

This work introduces ExtraFerm, an open-source simulator tailored to particle-number-conserving matchgates and controlled-phase gates, enabling efficient computation of Born-rule probabilities with runtime scales that are exponential only in circuit extent or CP-gate count. It provides three algorithms (Raw Estimate, Estimate, Exact) and trajectory-based techniques that make exact or approximate probability calculations tractable for large chemistry circuits, while offering LuCJ-specific optimizations for practical performance. The authors demonstrate substantial latency and memory advantages over tensor-network and state-vector baselines, and integrate ExtraFerm with SQD to significantly improve molecular ground-state energy estimates with minimal overhead, including real hardware demonstrations on nitrogen using the Heron processor. These results enable new hybrid quantum-classical workflows for near-term quantum chemistry and point to practical paths for leveraging matchgate-based circuits in quantum simulations.

Abstract

We present and open source a quantum circuit simulator tailored to chemistry applications. More specifically, our simulator can compute the Born-rule probabilities of samples obtained from circuits containing passive fermionic linear optical elements and controlled-phase gates. We support both approximate and exact calculation of probabilities, and for approximate probability calculation, our simulator's runtime is exponential only in the magnitudes of the circuit's controlled-phase gate angles. This makes our simulator useful for simulating certain systems that are beyond the reach of conventional state vector methods. We demonstrate our simulator's utility by simulating the local cluster unitary Jastrow (LUCJ) ansatz and integrating it with sample-based quantum diagonalization (SQD) to improve the accuracy of molecular ground-state energy estimates. Applied to a 52-qubit $N_2$ system, we observe accuracy improvements of up to $46\%$ over the baseline SQD implementation with negligible computational overhead. More generally, we highlight a regime in which our simulator achieves substantially superior latency scaling and exponentially superior memory scaling over a tensor network simulator and a state vector simulator. As an efficient and flexible tool for simulating quantum chemistry circuits, our simulator enables new opportunities for enhancing near-term quantum algorithms in chemistry and related domains.

Figures (11)

• Figure 1: The green star indicates the class of circuits that ExtraFerm is designed to address: circuits containing $m$ matchgates and $k$ controlled-phase gates where $k \ll m$. This work concentrates on particle number-conserving circuits. Note that matchgates + controlled-phase gates form a universal gate set.
• Figure 2: A high-level depiction of how ExtraFerm computes the Born-rule probability of a bitstring using Raw Estimate or Exact. The number of trajectories, $t$, can be given arbitrarily for Raw Estimate, while Exact sums over all possible trajectories. In practice, we parallelize this computation across bitstrings and then parallelize again across trajectories. Separate trajectories are generated for each bitstring to preserve the independence of Born-rule probability estimates.
• Figure 3: A visualization of how a trajectory $x$ and corresponding mode transformation matrix $V(x)$ are generated from a circuit. Each matchgate $m_i$ is unchanged while the controlled-phase gates are probabilistically assigned to either $d_0(\theta_j)$ or $d_1(\theta_j)$ based on $|\theta_j|$. In the above example, the trajectory $x = 010$ has been sampled. The first controlled-phase gate is assigned to $d_0(\theta_0)$ with probability $1$ while the second and third controlled-phase gates are assigned to $d_1(\theta_1)$ and $d_0(\theta_2)$, each with probability 1/2. Note that controlled-phase gates may operate on any two qubits; they are not restricted to nearest neighbors.
• Figure 4: Latency and memory comparisons of different tools when used to calculate the exact probability of an outcome measurement from a randomly generated particle number-conserving extended matchgate circuit. Each circuit contains 16 controlled-phase gates and a quantity of matchgates proportional to circuit size. Results are collected on an Intel Xeon w7-2495X processor. By default, all tools are given access to 24 cores, but only ExtraFerm benefits significantly from using more than 1. We also include results for ExtraFerm restricted to 1 core. Note that the y-axis for each graph is on the logarithmic scale (higher values are orders of magnitude worse).
• Figure 5: Speedup as a function of the number of cores used on a 24-core Intel Xeon w7-2495X processor for executing ExtraFerm on a 40-qubit LUCJ circuit containing 43 controlled-phase gates with angles sampled from $\mathcal{N}(0, 10^{-3})$ and a total extent 1.732. For each data point, Raw Estimate was used to compute probabilities for 1,000 bitstrings and times were averaged over 10 trials.