Variational Real-Time Dynamics on Reduced Operator Manifolds

AI summary

Overview

Research area: Quantum simulation of real-time dynamics for correlated quantum many-body systems, combining variational quantum algorithms with quantum embedding (ghost-Gutzwiller) and quantum-selected configuration interaction (QSCI).

Technical level: Intermediate to Advanced. The conceptual picture is accessible, but the theory section involves the Liouville–von Neumann equation, Ehrenfest equations, operator Jacobians and regularized least squares.

Scope (one sentence): The paper introduces and benchmarks Operator-Projected Variational Quantum Real-Time Evolution (OVQRTE), a variational algorithm that enforces the Ehrenfest equations for a chosen, polynomially sized set of observables instead of the full state evolution, and tests it on the Heisenberg model in simulation and on Anderson impurity models on superconducting hardware.

What This Paper Is About

Simulating how correlated quantum systems evolve in real time is hard both classically (the wavefunction is exponentially large) and on near-term quantum hardware (Trotterization and related methods need circuit depths that grow with simulated time and accuracy). Existing variational real-time evolution methods avoid deep circuits but pay for it by estimating the quantum geometric tensor, which requires many state-overlap measurements that scale poorly with the number of parameters.

The paper's goal is to obtain accurate short- and intermediate-time dynamics while measuring only ordinary expectation values. It does this by tracking a chosen subset of observables and forcing the variational parameters to reproduce the exact Ehrenfest equations for those observables, then reusing the resulting states as a source of configurations for ground-state estimation via QSCI inside a gGut embedding loop.

Key Contributions

1. A new algorithm, OVQRTE. The real-time counterpart of the imaginary-time OVQITE framework is derived: parameter velocities are fixed by a least-squares fit of the variational Ehrenfest residual, giving a linear system $A\dot{\bm\theta}=C$ with $A=G^{\top}G$, the Gram matrix of the observable Jacobian. This is explicitly not the quantum geometric tensor, so no Hadamard- or SWAP-test overlap circuits are needed.

2. A cost analysis showing a measurement advantage over VQRTE. The per-iteration circuit count is $C_{\text{total}} = 2p,C_{\text{pool}} + C_{\text{comm}}$, reducible to $2N_s C_{\text{pool}} + C_{\text{comm}}$ with SPSA, with $N_s$ typically 10–20 and not required to grow with parameter count or system size. Circuit counts scale linearly (or constant) in the number of parameters $p$, rather than quadratically as for quantum-geometric-tensor-based methods; a classical-shadow alternative with sample complexity $O(3^{\ell'}\log|\mathcal{S}|/\epsilon^2)$ at locality $\ell' = k+\ell-1$ is also noted.

3. A hardware-friendly, QEB-inspired operator pool. Jordan-Wigner parity strings of Pauli-$Z$ are replaced by identities, mapping $P_{ij}^{\mu\nu} = \sigma_i^\mu(\prod_{k=i+1}^{j-1} Z_k)\sigma_j^\nu$ to $\sigma_i^\mu \sigma_j^\nu$, shortening Pauli strings for measurement at the cost of being an approximate, qubit-excitation-inspired rather than algebraically exact reduction.

4. A demonstrated quantum–classical workflow for materials-style problems. OVQRTE trajectories are sampled in the computational basis to build cumulative configuration subspaces for QSCI, combined with self-consistent configuration recovery (S-CORE) and, for the half-filled Anderson impurity model, explicit closure of the recovered set under particle-hole symmetry, and wrapped inside a self-consistent gGut embedding loop.

Main Findings

• Large measurement reduction on the Heisenberg model. For the open-boundary 1d Heisenberg model with $J=1/2$ and transverse field $h=-1$, using the two-local nearest-neighbor Pauli set $\mathcal{S}_{2\text{NN}}$ and a hardware-efficient ansatz with $L=1$, OVQRTE requires roughly one to two orders of magnitude fewer measurement circuits per time step than VQRTE. The advantage grows with system size, with the SPSA variant approaching close to two orders of magnitude reduction for the largest systems considered (16, 32 and 50 qubits in Fig. 1).

• Exact dynamics in the statevector limit. For a 12-qubit chain, projecting onto $\mathcal{S}_{2\text{NN}}$ reproduces the exact trajectory to numerical precision, with infidelity $1-F < 10^{-9}$, for both OVQRTE and VQRTE.

• Better finite-shot noise resilience. With sampling, OVQRTE shows a late-time infidelity plateau of $1-F \sim 10^{-2}$ at $N_{\text{sh}} = 10^{4}$ shots and $1-F \sim 10^{-3}$ at $N_{\text{sh}} = 10^{5}$. VQRTE saturates roughly one order of magnitude higher at each shot budget, so OVQRTE at $10^4$ shots reaches the fidelity VQRTE attains only at $10^5$.

• Sharper separation in observable errors. The transverse magnetization $M_X = \frac{1}{n}\sum_i \langle X_i\rangle$ shows VQRTE's pointwise error growing to $|\Delta M_X| \sim 0.75$ at $N_{\text{sh}} = 10^{4}$, while OVQRTE's error stays an order of magnitude smaller across both shot budgets. The conserved longitudinal magnetization $M_Z$ (exactly zero for this evolution) gives $|\Delta M_Z| \lesssim 0.2$, versus $|\Delta M_X| \lesssim 0.75$ for the VQRTE comparison.

• Hardware benchmark on Anderson impurity models. The algorithm was benchmarked for Anderson impurity models on the IQM Emerald superconducting processor using up to 24 qubits, with a LUCJ ansatz at $L=1$ and the $\mathcal{S}_{1+ZZ}$ operator set. (Specific hardware accuracy and energy values are not reported in the truncated text supplied.)

• Ground-state energy and density of states via QSCI + gGut. OVQRTE-generated computational-basis samples were used to build QSCI subspaces, yielding ground-state energy and density of states inside the self-consistent gGut loop. The authors state that full trajectory accuracy is not required for this use—only that the evolved states place appreciable weight on configurations spanning an informative low-energy subspace.

• Two independently removable approximations. The method rests on (i) information restriction to the tracked operator set and (ii) the variational realization on the operator manifold $\mathcal{M}_{\text{op}}$; enlarging $\mathcal{S}$ toward a complete basis recovers full McLachlan evolution, and enlarging the ansatz toward universality removes the second.

Methodology in Plain English

The starting point is that a quantum state is fully described by the expectation values of a complete set of observables, but that set is exponentially large ($\mathcal{O}(4^n)$ for $n$ qubits). The authors instead pick a polynomial-size subset—here, single Pauli operators plus nearest-neighbor Pauli pairs, or the operators defining one- and two-body reduced density matrices—chosen by locality, since commuting a local operator with a local Hamiltonian raises its weight only by the Hamiltonian's own weight.

For each tracked observable, the exact quantum mechanical rate of change is given by the Ehrenfest equation, $i\langle[\hat H,\hat O_\alpha]\rangle$. The rate of change that the parameterized circuit can actually produce is a linear combination of parameter velocities weighted by the gradient of each observable with respect to each parameter. OVQRTE chooses the parameter velocities that best match the two in a least-squares sense, which turns into a small linear system solved with a regularized least-squares solver to guard against ill-conditioning, followed by a simple forward-Euler step in time. All quantities entering the system are ordinary expectation values measured on the current state, grouped into qubit-wise commuting sets so that many operators are read out from the same circuit.

Parameter gradients are obtained either by finite differences (two circuit settings per parameter) or by simultaneous perturbation stochastic approximation (SPSA), which perturbs all parameters along one random direction and needs only two evaluations per sample. To reduce noise from Jordan-Wigner parity strings, the operator pool drops intermediate Pauli-$Z$ strings in a qubit-excitation-inspired simplification.

For the downstream spectral calculation, the time-evolved states are measured repeatedly in the computational basis; the union of all bitstrings seen up to each iteration defines a configuration subspace, which is then used to project and diagonalize the Hamiltonian. Noisy samples that fall outside the conserved particle-number sector are not discarded but probabilistically corrected with S-CORE using average spin-orbital occupations from the current CI solution; for the half-filled Anderson model the set is additionally closed under particle-hole symmetry. This solver is embedded in the self-consistent gGut loop, where the impurity ground state and one-particle reduced density matrix must be refreshed each iteration.

Why This Matters

Impact on research. The work shows that a variational real-time algorithm can be built without the quantum geometric tensor, replacing an $\mathcal{O}(p^2)$ overlap-estimation bottleneck with $\mathcal{O}(p)$ (or constant, under SPSA) expectation-value circuits. It also strengthens the case that real-time evolution is valuable even when the trajectory is only approximate, because the resulting states are useful generators of compact subspaces for ground-state and spectral calculations—directly connecting dynamics, QSCI, and quantum embedding.

Real-world applications:

• Materials simulation of correlated electron systems, where dynamical response functions and densities of states are the target quantities.
• Strongly correlated impurity problems arising in quantum embedding workflows (DMFT, DMET, gGut) used for transition-metal compounds and molecular complexes.
• Quantum chemistry ground-state energy estimation through selected configuration interaction seeded by quantum sampling.
• Near-term hardware deployments that are measurement-limited rather than gate-limited, where reducing circuit counts per time step is the binding constraint.

Industry relevance. The demonstration uses up to 24 qubits on the IQM Emerald superconducting processor and is co-authored from IQM Quantum Computers, indicating a direct pathway toward commercially relevant quantum-assisted materials workflows with modest qubit counts. The reduction in measurement overhead per time step is a practical metric for cloud-billed quantum compute time.

Future Directions

• Determining how the choice of tracked operator set (size, weight, and whether it is built from Pauli strings or RDM operators) controls long-time trajectory accuracy, given that exact invariance of the set under the Hamiltonian commutator is exponentially costly and that untracked components can drift and feed back into tracked derivatives.
• Quantifying systematically the trade-off between the two approximation knobs—operator-set size and ansatz expressivity—including whether enlarged sets meaningfully improve dynamics at fixed measurement budget.
• Scaling the QSCI-plus-gGut pipeline to larger impurity problems and multi-band models, where the qubit counts exceed the 24-qubit hardware demonstration and where symmetry completion beyond particle-hole symmetry may be needed.
• Assessing how far the QEB-inspired replacement of Pauli-$Z$ parity strings by identities can be pushed without loss of accuracy, and how that interacts with noise and with the S-CORE error-mitigation layer.

Target Audience

Quantum-algorithm researchers working on variational simulation and near-term quantum dynamics; quantum-embedding and materials-simulation practitioners interested in quantum impurity solvers; and quantum-computing hardware teams evaluating measurement-cost budgets for real-device workloads. Readers should be comfortable with variational quantum circuits, Pauli operator algebra, and basic quantum many-body notation to get the most from the theory and results sections.

Authors’ abstract

Accurate real-time simulation of correlated quantum systems remains challenging for both classical methods and near-term quantum hardware. We introduce operator-projected variational quantum real-time evolution (OVQRTE), which updates a parameterized circuit by enforcing the Ehrenfest equations for a selected set of observables. OVQRTE requires only expectation-value measurements, while the choice of operator set enables a systematic trade-off between accuracy and measurement cost, substantially reducing quantum-resource requirements relative to existing variational real-time-evolution algorithms. After implementing OVQRTE dynamics of Heisenberg model on a simulator, we benchmark the algorithm for the Anderson impurity models on the IQM Emerald superconducting processor using up to 24 qubits. We further use OVQRTE to sample computational-basis states for quantum-selected configuration interaction (QSCI), enabling the calculation of the ground-state energy and density of states within a self-consistent ghost-Gutzwiller Ansatz (gGut) embedding loop. Our results establish OVQRTE as a promising approach for investigating correlated condensed-matter systems on near-term quantum hardware.