Unitary imaginary time evolution and ground state preparation using multi-copy protocols

Abstract

Efficient low-energy state preparation is a key objective in quantum computation and quantum simulation. Quantum imaginary-time evolution replaces real-time dynamics with imaginary-time dynamics, exponentially suppressing higher-energy eigenstates. We introduce deterministic unitary protocols that approximate imaginary-time evolution for ground-state preparation. The protocols require multiple copies of the system, real-time evolution under the system Hamiltonian, and controlled-SWAP operations (or more general SWAP-generated unitaries). We analyze two concrete circuit families: a tree architecture with provable polynomial-in-depth convergence but rapidly growing width, and a compact "hedge" architecture that achieves comparable accuracy with only polynomial width in a heuristic construction supported by numerics. We provide numerical evidence that mid-circuit post-selection can accelerate convergence with practical success probabilities. Separately, we demonstrate that circuit volume can be traded for the shot complexity of post-circuit observable estimation in the ground-state preparation setting. We outline concrete implementation of platform-specific routes, where multi-copy registers and SWAP-mediated couplings are natural, thereby illustrating how these hybrid analog-digital circuits can complement existing state-preparation methods in the near term.

Figures (5)

• Figure 1: $(a)$ One small step of unitary imaginary time evolution $\propto \epsilon$ w.r.t. the target Hamiltonian $H$ "heats" or "cools" the state of an individual copy. $(b)$ We employ a unitary gate that can approximately evolve simultaneously a pair of copies forward and backward in imaginary time when applied to a pair that is symmetric under SWAP. $(c)$ The circuit illustrates how identically initialized copies can be re-used to collectively propagate multiple steps forward and backward in imaginary time. $(d)$ In an idealized implementation of our protocols the first(last) copy evolves many steps $n$ forward(backward) in imaginary time $\beta$, thus asymptotically preparing a state of minimal(maximal) energy $\langle H \rangle$.
• Figure 2: (a) The tree circuit for $8$ copies. (b) The hedge circuit for $8$ copies. To advance the first copy $n$ steps, the tree circuit uses $2^n$ copies and $2^n-1$ gates, while the hedge circuit uses $2n$ copies and $\sim n^3$ gates.
• Figure 3: Numerical results for the ground state preparation of the mixed field Ising Hamiltonian $H=-\sum_{i}( \sigma_z^{i}\sigma_z^{i+1}+\sigma_z^{i} + \sigma_x^{i})$ with periodic boundary conditions and $N$ spins, using the tree circuit in Sec. \ref{['exponential']}. The initial pure state is defined as $\ket{\phi_0} = \sqrt{p}\ket{\phi_{gs}}+\sqrt{1-p}\ket{\phi_{\perp}}$, where $\ket{\phi_{\perp}}$ is a normalized equal weight superposition of all energy eigenvectors, except the ground state, $\ket{\phi_{gs}}$. The ground state energy is $E_{gs}$. All data points are given by minimizing the protocol's output energy, $\tilde{E}$, over $\epsilon$. $(a)$ Relative reduction in energy, $(E_{gs}-\tilde{E})/(E_{gs}-E_0)$ as a function of $p$, for $n=10$ layers. Here, $E_0$ is the energy expectation value of the protocol's input state. $(b)$$(E_{gs}-\tilde{E})/(E_{gs})$ as a function of the number of layers with $p=1/2^N$. $(c)$ The minimal $n$ needed for $(E_{gs}-\tilde{E})/(E_{gs})\leq 0.05$, $n_*$, as a function of $N$. $p=1/2^N$ represents an input state with a random-like behavior, and $p=0.5$ represents an input state that has been prepared to be close to the ground state.
• Figure 4: Simulations of the protocol with the hedge circuit in Sec. \ref{['hedge']} with $H=\sigma_z$, The initial state is given by $2n$ copies of the $-\sigma_x$ ground state. (a) MPS simulation using bond dimension $\chi =400$ for the fidelity with the ground state, as a function of $n \epsilon$. The dashed curve represents the exact imaginary time evolution with imaginary time $n\epsilon$. (b) MPS simulation for the infidelity with the exact ground state as a function of the number of copies, where the protocol is optimized over the step parameter $\epsilon$ to give minimal energy. $\chi$ is the bond dimension. (c) Comparison of the Hedge circuit performance for ground state preparation, given by the distance to the ground state energy, with and without post-selection. The solid line is the total probability of the post-selection. Both protocols are optimized over the step parameter $\epsilon$ to give minimal energy. (d) Infidelity with the exact imaginary evolved state as a function of $n$, normalized to $1$ at $n=1$. The infidelity at $n=1$ for the different increasing values of $\beta$ is $1-\mathcal{F}_\beta^{(1)} = (0.39 \cdot 10^{-3},0.26,0.27,0.50,0.25,0.58)$.
• Figure 5: Numerical results for the single-layer protocol. $E_{gs}-\tilde{E}$ is the difference between the ground state energy and ${\rm Tr}(H_{int} \tilde{\rho}^n)/{\rm Tr}( \tilde{\rho}^n)$, where $H_{int} = -\sum_{i}(\sigma_z^i \sigma_z^{i+1}+\sigma_x^{i}+\sigma_z^{i})$, is a spin chain Hamiltonian of $N$ spins with periodic boundary conditions. $\Delta_N$ is the energy gap between the ground and first excited state. $\epsilon_N$ is taken to be $0.19/N$, for one data set, and $0.03/N$ for the other. The results show that ${\rm Tr}(H_{int} \tilde{\rho}^n)/{\rm Tr}( \tilde{\rho}^n)\approx E_{gs}(1+\Delta_N e^{-a n })$. When $\epsilon$ is small enough, $a = 2 n \epsilon \Delta_N$, as $\tilde{\rho} \approx e^{-2 \epsilon H_{int}}$ approximates the thermal state. Increasing $\epsilon$ breaks the thermal state approximation, but can yield a faster apparent decay (larger fitted $a$).