Source-linked AI summary
Variational ansatz-based quantum simulation of imaginary time evolution
Sam McArdle, Tyson Jones, Suguru Endo, Ying Li, Simon Benjamin, Xiao Yuan
TL;DR
Imaginary-time evolution is difficult to implement on quantum computers because it is non-unitary, while classical many-body simulation generally scales exponentially with system size. The paper introduces a hybrid quantum-classical variational algorithm and finds it outperforms gradient descent for ground-state energy tasks on H2 and LiH.
Problem
Non-unitary imaginary-time evolution cannot be naively mapped to quantum circuits, while classical many-body simulation generally scales exponentially with system size.
Method
The paper projects imaginary-time evolution onto a parameterized ansatz manifold and updates its parameters with a hybrid quantum-classical variational algorithm.
Results
Imaginary-time evolution outperformed gradient descent for finding ground-state energies, with tests on H2 and LiH.
Takeaways & Limitations
The method provides a competitive alternative to conventional classical optimisation routines for ground-state energy tasks.
Takeaways & Limitations
The variational method may converge to higher-energy states when the chosen ansatz is not sufficiently powerful.
Abstract
from arXiv · showhide
Imaginary time evolution is a powerful tool for studying quantum systems. While it is possible to simulate with a classical computer, the time and memory requirements generally scale exponentially with the system size. Conversely, quantum computers can efficiently simulate quantum systems, but not non-unitary imaginary time evolution. We propose a variational algorithm for simulating imaginary time evolution on a hybrid quantum computer. We use this algorithm to find the ground-state energy of many-particle systems; specifically molecular hydrogen and lithium hydride, finding the ground state with high probability. Our method can also be applied to general optimisation problems and quantum machine learning. As our algorithm is hybrid, suitable for error mitigation and can exploit shallow quantum circuits, it can be implemented with current quantum computers.
INTRODUCTION
Imaginary time evolution is difficult to realize on quantum computers because its operator is non-unitary, despite quantum computers’ suitability for representing many-body states. The paper proposes a hybrid quantum-classical variational method to simulate imaginary time evolution and target many-body ground-state energies with low-depth circuits.
- Imaginary time is a powerful mathematical concept used in quantum mechanics, statistical mechanics, and cosmology.
- Classical many-body quantum simulation is generally hard because wavefunction dimension grows exponentially with particle number.Classical approaches include propagator evaluation, quantum Monte Carlo, and density matrix renormalization group methods.
- Quantum computers efficiently represent many-body states and can simulate real-time evolution through unitary quantum circuits.Trotterization decomposes the real-time propagator into single- and two-qubit gates.
- Imaginary-time evolution is not straightforward to implement with quantum circuits because its operator is non-unitary and cannot be directly decomposed through Trotterization.Alternative methods are therefore required for quantum implementation.
- The paper proposes a hybrid quantum-classical variational algorithm that combines quantum many-body state representation with classical simulation of arbitrary processes.The method is discussed for finding ground-state energies of many-body quantum systems and requires only a low-depth circuit for current and near-term processors.
RESULTS
Variational imaginary-time evolution robustly finds low-energy states and can outperform gradient descent, especially on challenging or randomly initialized landscapes. The method also shows resilience to noise, while its effectiveness depends on ansatz expressivity and sufficiently accurate gradient measurements.
- Limitations: The ansatz may fail to faithfully represent every state along the desired trajectory, despite promising numerical results for imaginary-time evolution.The authors describe the method as a robust routine for energy minimization, while noting that ansatz coverage is difficult to prove.
- Energy landscapes: Imaginary time robustly discovered system A’s global minima, whereas gradient descent became trapped in local minima.For system B, imaginary time performed comparably to gradient descent.
- Convergence: 0.225 versus 0.886: imaginary-time evolution converged faster than gradient descent and located the ground state more accurately.The advantage was most noticeable for random start states.
- Convergence: The obtained convergence rate was significantly higher than gradient descent, supporting applications to optimization problems with difficult initial states.The passage identifies QAOA as a potentially relevant application where good starting states are harder to motivate.
- Noise robustness: Imaginary time performed significantly better than gradient descent under noise, even when substantially fewer measurements were made.If NC < 2 × 10^4, increasing NA could not improve reliability and could make imaginary time less effective than gradient descent.
DISCUSSION
The authors propose a hybrid quantum-classical method for imaginary-time evolution and report applications to ground-state energy estimation in H2 and LiH. They argue that the method also supports optimization and quantum machine learning, while noting convergence limitations from insufficiently expressive ansätze.
- Applications: The method was tested for ground-state energy estimation in H2 and LiH.The discussion identifies these systems as applications of the proposed hybrid quantum-classical imaginary-time simulation method.
- Optimization: Imaginary-time evolution outperformed gradient descent for this task, suggesting a competitive alternative to conventional classical optimization routines.The authors propose further study of the method for general optimization problems, including in conjunction with QAOA.
- Quantum machine learning: The method can prepare thermal Gibbs states and may therefore apply to quantum machine-learning problems involving Gibbs-distribution sampling.The Gibbs state is defined as ρT = e−H/T /Tr[e−H/T ].
- Quantum machine learning: Unlike previous Gibbs-state preparation methods requiring long gate sequences and fault tolerance, this method can use a shallow circuit.The discussion links shallow-circuit implementation to the method’s suitability for current quantum-computing applications.
- Limitations and future work: The variational method may converge to higher-energy states when the chosen ansatz is insufficiently powerful.Future work will investigate applications across chemistry, optimization, and machine learning, including suitable ansätze for different problems.
Supplementary Information
The supplementary information derives the variational imaginary-time equations, shows that the average energy decreases for sufficiently small stepsizes, and describes quantum-circuit evaluation of the required coefficients. It also details molecular reductions and regularization used in simulations.
- Variational principle: McLachlan’s variational principle projects imaginary-time evolution onto the subspace reachable by the parameterized ansatz.Replacing the general state with |φ(θ1, θ2, . . . , θN)⟩ yields the projected evolution principle.
- Variational principle: The variational condition is equivalent to a differential equation governing the ansatz parameters.The parameter dynamics follow from McLachlan’s variational requirement.
- Energy behavior: For a sufficiently small stepsize, the algorithm’s average energy always decreases along the imaginary-time evolution.The energy is defined as E(τ) = ⟨φ(τ)| H |φ(τ)⟩.
- Quantum-circuit implementation: The coefficients A and C can be efficiently evaluated with quantum circuits using derivatives of rotation and controlled-rotation gates.The derivative terms reduce to measurable circuit expressions, and the full controlled-U gate is unnecessary in practice.
- Molecular simulation: The H2 Hamiltonian is reduced from a four-qubit Bravyi-Kitaev form to a Hamiltonian acting on two qubits.The reduced Hamiltonian contains I, Z0, Z1, Z0Z1, Y0Y1, and X0X1 terms.
- Regularization: 10^-4–10^-2: the Tikhonov parameter is constrained to this range while balancing accuracy, smooth parameter evolution, and energy monotonicity.The ideal value is estimated from a 3-point L-curve; λ = 0 allowed residuals in A^-1 to disrupt monotonic energy decrease, while excessive regularization could converge to non-ground states.