Subspace Variational Quantum Simulation: Fidelity Lower Bounds as Measures of Training Success

TL;DR

This work develops Subspace Variational Quantum Simulation, an iterative variational protocol that compresses Trotter time evolution for a chosen subspace into a fixed-depth parameterized circuit trained over multiple initial states. A fidelity-based cost function, augmented with pairwise superposition states, ensures correct subspace dynamics, while a computable fidelity lower bound is obtained via SDP relaxation of a QCQP to guarantee worst-case performance. The authors prove a warm-start region free of barren plateaus for multi-state training and demonstrate the approach experimentally on a 2-qubit Ising model and numerically on a 10-qubit Ising model, showing robust performance and scalability with expressive PQCs. These results offer an efficient, trainable framework for subspace quantum dynamics and provide a practical fidelity bound that supports reliable operation on near-term devices and potential Krylov-subspace applications.

Abstract

We propose an iterative variational quantum algorithm to simulate the time evolution of arbitrary initial states within a given subspace. The algorithm compresses the Trotter circuit into a shorter-depth parameterized circuit, which is optimized simultaneously over multiple initial states in a single training process using fidelity-based cost functions. After the whole training procedure, we provide an efficiently computable lower bound on the fidelities for arbitrary states within the subspace, which guarantees the performance of the algorithm in the worst-case training scenario. We also show our cost function exhibits a barren-plateau-free region near the initial parameters at each iteration in the training landscape. The experimental demonstration of the algorithm is presented through the simulation of a 2-qubit Ising model on an IBMQ processor. As a demonstration for a larger system, a simulation of a 10-qubit Ising model is also provided.

Figures (8)

• Figure 1: Schematic diagram of the algorithm. Each numbered box corresponds to the matching step in the algorithm described above. 1. Preparation of the initial states. 2. The process of the $m$-th iteration. The cost function is evaluated by the quantum circuit shown, via fidelity measurements. The folded gates ${V_i}$ prepare the initial states such that $|\Psi_i\rangle = V_i|\bm{0}\rangle$ (similarly for $|\Psi_i^+\rangle$). Based on the cost function value, the classical computer (navy monitor) updates the variational parameters according to a chosen update rule (e.g., gradient descent), and the cost function is minimized iteratively. 3. After completing the iteration for $m = 1$ to $n$, the optimized parameters ${\bar{\bm{\phi}}_m}$ are returned as the output.
• Figure 2: SDP-based calculation of the fidelity lower bound. (a) and (b) are the fidelity lower bound for $\cos(\theta/2)|\Psi_0\rangle+e^{i\phi}\sin(\theta/2)|\Psi_1\rangle$ state, under the constraints $F_0=F_1=F_1^+=0.99$ and $F_0=F_1=F_1^+=F_1^-=0.99$, respectively. (c) and (d) show the fidelity lower bound for the state $\cos(\pi/4)|\Psi_0\rangle+e^{i\phi}\sin(\pi/4)|\Psi_1\rangle$ as a function of the perturbation $\epsilon$, given as the infidelity $F=1-\epsilon$. (c) uses constraints $F_0=F_1=F_1^+= 1-\epsilon$, based on three initial states: $|\Psi_0\rangle, |\Psi_1\rangle$, and $|\Psi_1^+\rangle$. (d) includes an additional fidelity constraint $F_1^-= 1-\epsilon$ for the initial state $|\Psi_1^-\rangle$.
• Figure 3: PQCs used in the experiment. The $R$ gate denotes an Euler rotation with three parameters, and the $R_i$ gate is a single-qubit rotation about the $i$-axis. (a) An ansatz called SU(4), which represents all possible $4 \times 4$ unitary matrices up to a global phase, with 15 variational parameters. (b) A double-layered ansatz consisting of single-qubit rotations along the $z$, $x$, and $z$ axes, followed by CNOT gates, referred to as the $ZXZ$+CNOT ansatz.
• Figure 4: Fidelities between the states evolved under the Trotter circuit and the parameterized states produced by the trained PQC. Solid lines show the fidelities of the three training target states, while dotted lines indicate their corresponding lower bounds from Proposition \ref{['prop:lb_m']}. Transparent yellow lines represent the fidelities of 500 randomly sampled initial states within the subspace. (a) shows the results with the SU(4) ansatz, and (b) those with the ZXZ+CNOT ansatz.
• Figure 5: Quantum simulation of the dynamics of concentratable entanglement, for multiple initial states $|\psi_0(\theta)\rangle$. Time evolution is implemented using (a) a pre-trained PQC with a SU(4) ansatz, (b) a second-order Trotter circuit, and (c) the ideal time evolution operator. Simulations in (a) and (b) are carried out on the quantum processor, while (c) is obtained by numerical calculation.

Theorems & Definitions (8)

• Proposition 1
• proof
• Theorem 1
• proof
• Proposition 2
• proof
• Theorem 2
• proof