Imaginary time evolution of a quantum system through analytic continuation from real-time quantum simulation

AI summary

Overview

Research area: Quantum simulation algorithms — specifically methods for implementing imaginary-time evolution (ITE) on quantum computers, with connections to quantum Monte Carlo, spectral filtering, and eigenstate preparation.

Technical level: Advanced. The paper relies on complex contour integration, Green's function spectral representations, Trotterized quantum circuits, and Hadamard-test measurement protocols.

Scope (one sentence): The paper derives and demonstrates that the non-unitary imaginary-time evolution operator can be reconstructed by analytic continuation of ordinary real-time correlation functions measured on a quantum computer, using a single free parameter — a lower bound on the ground-state energy — and validates the method on solvable 1D diffusion and scattering problems plus small-scale IBM hardware runs.

What This Paper Is About

Real-time evolution on a quantum computer is straightforward because the operator is unitary and can be Trotterized, but imaginary-time evolution is non-unitary and therefore much harder to implement directly, even though it is valuable both for systems where classical Monte Carlo suffers sign problems and as an exact route to ground-state preparation. The authors ask whether ITE can instead be obtained after the fact, by measuring ordinary real-time correlation functions at a finite set of times and analytically continuing them to imaginary time. They test this idea on exactly solvable 1D problems and on IBM quantum hardware.

Key Contributions

1. An exact integral identity linking ITE to real-time evolution. For a Hermitian Hamiltonian with a lower bound c on the ground-state energy, the paper proves e^{-Ĥτ} = (i/2π) ∫_{-∞}^{+∞} dt [e^{ic(t+iτ)}/(t+iτ)] e^{-iĤt} (Eq. 1/2). The identity follows from closing the t-contour in the negative imaginary half-plane and picking up the residue at t = -iτ.

2. A controlled truncation-error analysis. The infinite integral is cut to ∫_{-t_cut}^{+t_cut} dt, and a relative error bound is derived for t_cut > τ: at most (2/π)·(t_cut+τ)/(t_cut²+τ²)·e^{(E-c)τ}/(E-c) (Eq. 3), giving 1/t_cut convergence for fixed c. The practical integral is then evaluated by Gaussian quadrature at finitely many times t_n, n ∈ {0, …, N_samples}.

3. An independent derivation by contour deformation. Starting from the Green's-function representation e^{-Ĥτ} = (1/2π) ∫_C dE e^{-Eτ}/(E-Ĥ) with C the vertical line c - i∞ to c + i∞ and c < E₀, the contour is deformed into four pieces (with a positive parameter a), and the identity -i∫₀^∞ dt e^{iEt} e^{-iĤt} = 1/(E-Ĥ) (valid for Im[E] > 0) converts each piece into a real-time evolution operator. Performing the resulting double integrals in ε recovers the single-integral result of Eq. 2 (Eq. 14 is the intermediate form).

4. Three worked applications and small-scale hardware demonstrations. The method is applied to the 1D Fokker–Planck equation for Brownian motion with constant drift, the Ornstein–Uhlenbeck process, and imaginary-time evolution of the integrated correlation function in 1D contact-interaction scattering; numerical tests on 1 and 2 qubits are run on IBM hardware and compared to exact solutions.

Main Findings

• The integral acts as a spectral filter. The right-hand side of the master formula implements the filter e^{-Eτ} Θ(E-c), where Θ is the Heaviside step function. If an eigenenergy E < c exists (i.e. c was chosen badly), the contour closes in the upper half-plane and the integral evaluates to exactly 0 — the state is filtered out rather than represented incorrectly.

• Choice of c is a two-sided trade-off. Choosing c tightly improves convergence by suppressing the exponential factor e^{(E-c)τ}, but setting c too close to the ground-state energy E₀ causes a 1/(E-c) blow-up in the relative error. The paper reports this is confirmed both analytically and numerically.

• The 2×2 matrix test validates the identity. For Ĥ = [[E₀, g],[g*, E₁]] with E₀ = 0.6, g = 0.2 + i·1.1, E₁ = 2.3, a = 1, c = 0, the analytic-continuation results match e^{-Ĥτ} at t_cut = 50 and t_cut = 250; the t_cut = 50 curve shows some systematic error that is largely removed at t_cut = 250. The large-τ limit ⟨x|e^{-Ĥτ}|x⟩ → |⟨x|E₋⟩|² e^{-E₋τ} is also compared against the analytic-continuation result.

• Diffusion processes are reproduced accurately. For Brownian motion with constant drift (γ = 4.6, D = 1.5, a_x = 0.1, a = 1.0, c = 0) and the Ornstein–Uhlenbeck process (c = -0.2), using an 81×81 Hamiltonian matrix, t_cut = 100, and a rectangular initial condition of width b = 0.9, the recovered probability distributions W(x,τ) agree excellently with the exact error-function solutions of Eqs. 27 and 28. These runs were performed on classical computers by matrix computation.

• Scattering results agree at small τ and degrade at large τ. For ΔC(τ) = C(τ) - C₀(τ) with V₀ = 2, a_x = 0.1, a = 1.0, c = -0.1, t_cut = 100, and an 81×81 matrix, the analytic-continuation result matches the exact expression of Eq. 35 well for small t; deviations at large t are attributed primarily to the finite t_cut and to boundary-condition effects, and the authors state the agreement would improve with larger t_cut and a larger Hamiltonian matrix.

• Hardware implementation uses standard primitives. The Hamiltonian is split as Ĥ = Ĥ_a + Ĥ_b + Ĥ_v, with Ĥ_a and Ĥ_b matching the first two terms of a tight-binding model (circuits given for e^{-iĤ_a δt} and e^{-iĤ_b δt}, including periodic boundary conditions |0⟩ = |2^Γ⟩), Ĥ_v implemented as a diagonal unitary via the Qiskit operator class, and the full evolution Trotterized as [e^{-iĤ_v δt} e^{-iĤ_b δt} e^{-iĤ_a δt}]^{N_t} with t = δt·N_t. Real and imaginary parts of ⟨ψ|e^{-iĤt}|ψ⟩ are read from Hadamard-test ancilla probabilities as P(0)-P(1) and P(1)-P(0) respectively.

• Matrix elements need only two state preparations in this setting. A general matrix element ⟨α|e^{-iĤt}|e^{Φ/2}W(0)⟩ requires four expectation values built from |ψ_±^{(X)}⟩ = (|α⟩ ± |e^{Φ/2}W(0)⟩)/√2 and |ψ_±^{(Y)}⟩ = (|α⟩ ± i|e^{Φ/2}W(0)⟩)/√2; because the Hamiltonians here are completely real, the matrix element is transpose symmetric and the two Y terms cancel, leaving only the two X expectation values.

• Not reported in the supplied text: the quantitative hardware results for the 1- and 2-qubit IBM experiments (e.g. measured values, fidelities, or error metrics). The provided content is truncated within Sec. IV.1 after the definition of the normalized Brownian-motion initial state, so the specific hardware numbers are not available here.

Methodology in Plain English

The authors exploit a mathematical trick: instead of trying to build the non-unitary operator that shrinks a quantum state toward its ground state, they express it as a weighted sum (an integral) of ordinary, unitary real-time evolutions. Because a quantum computer can already run real-time evolution, the recipe becomes: run the circuit at a set of chosen times, measure the correlation functions, then combine those measured numbers with known complex weights on a classical computer.

The weights contain a single tunable ingredient, the number c, which must sit below the true ground-state energy. Its role is to guarantee that the integral picks up only the physically meaningful part of the spectrum — states below c are filtered to zero rather than corrupted. In practice the infinite time integral is cut off at ±t_cut and evaluated with Gaussian quadrature, and the authors derive an explicit bound showing the error shrinks like 1/t_cut.

To test the recipe, they use three 1D problems with known closed-form answers: two diffusion processes (Brownian motion with constant drift, and the Ornstein–Uhlenbeck process), both mapped onto Schrödinger-like equations with an effective potential, and a quantum scattering problem where the imaginary-time integrated correlation function encodes the scattering phase shift. All three are discretized on a lattice, evolved in real time, analytically continued, and compared against exact formulas. Finally, they map the lattice Hamiltonians onto qubit circuits — a tight-binding-style hopping part plus a diagonal potential part — Trotterize, and use a Hadamard test to extract the needed matrix elements on IBM hardware for 1 and 2 qubits.

Why This Matters

The work reframes ITE as a classical post-processing step on data that quantum computers are already good at producing. Because real-time evolution avoids the non-unitarity problem entirely, the approach sidesteps ancilla overhead and post-selection costs associated with embedding methods, and the variational approximations inherent in QITE/VarQITE. It also provides a clean spectral-filter interpretation, connecting quantum simulation to the filter-based eigenstate-preparation literature.

Real-world applications:

• Ground-state preparation for quantum chemistry and materials problems, where preparing eigenstates of generic Hamiltonians is otherwise difficult.
• Lattice field theory and sign-problem systems, where classical Monte Carlo becomes prohibitively expensive as system size grows but real-time quantum simulation remains viable.
• Stochastic and diffusion modeling — the Fokker–Planck demonstrations map directly onto classical Brownian motion and Ornstein–Uhlenbeck processes used in physics, finance, and biophysics.
• Nuclear and scattering physics — extracting scattering phase shifts from imaginary-time correlation functions is a standard tool, here made accessible through real-time simulation.

Industry relevance: the algorithm requires only Trotterized real-time evolution and Hadamard-test measurements, both standard on today's superconducting hardware, and the authors demonstrate it using IBM devices and Qiskit. That makes it a near-term-compatible protocol rather than a fault-tolerant-only construction, relevant to teams building quantum simulation stacks for chemistry, materials, and physics.

Future Directions

• Better filters for eigenstate preparation. The paper notes that given additional information about the low-lying spectrum and the overlap of the trial state onto the eigenstate of interest, more optimal filters than the pure e^{-Eτ} Θ(E-c) filter can be designed.
• How to obtain c in practice. The authors list three routes — classical Monte-Carlo bounds, explicit analytic bounds, or dedicated quantum algorithms — but the reliability and cost of each for large systems remains an open question, especially given the 1/(E-c) sensitivity when c approaches E₀.
• Scaling the time cutoff and lattice size. The scattering comparison degrades at large τ due to finite t_cut and boundary conditions; the authors state that larger t_cut and larger Hamiltonian matrices would improve agreement, leaving the resource scaling for realistic problem sizes to be characterized.
• Behavior on real hardware beyond 1–2 qubits. The reported tests are on 1 and 2 qubits; how Trotter error, gate noise, and the many-time sampling needed for Gaussian quadrature compound at larger qubit counts is the natural next question.

Target Audience

Quantum-algorithm researchers and graduate students working on quantum simulation, spectral filtering, and eigenstate preparation; computational physicists interested in sign-problem-free alternatives to classical Monte Carlo; nuclear and scattering theorists who already use imaginary-time correlation functions; and quantum-software practitioners on superconducting platforms looking for near-term-compatible protocols built from Trotterization and Hadamard tests. Readers should be comfortable with complex analysis and second-quantized/lattice Hamiltonians.

Authors’ abstract

Though quantum computing naturally offers an advantage for simulations of real-time quantum systems, implementing Imaginary-Time Evolution (ITE) is comparatively more difficult. Nevertheless, quantum implementations of ITE are useful both in cases where classical implementations have associated sign problems, and also as an exact method of preparing eigenstates on quantum computers. In this work we present an algorithm to obtain ITE of a generic quantum Hamiltonian by performing analytic continuation of measured real-time correlation functions. We present simulations and demonstrations on IBM quantum hardware of 1D Fokker-Planck equations for classical diffusion process and imaginary-time evolution of integrated correlation functions in 1D quantum mechanical scattering as examples to demonstrate the effectiveness of the method.