Double-bracket quantum algorithms for quantum imaginary-time evolution

TL;DR

<3-5 sentence high-level summary> DB-QITE introduces a coherent quantum-circuit realization of imaginary-time evolution by encoding Brockett's double-bracket flow into unitary operations. The algorithm inherits the cooling properties of imaginary-time evolution, providing energy reduction tied to energy fluctuations and exponential fidelity improvement with iteration, while avoiding classical optimization or block-encodings. Numerical simulations on Heisenberg models show competitive gate counts and high fidelities, with warm-starting via DB-QITE enhancing QPE performance under practical hardware constraints. The work includes a detailed runtime analysis and a thorough comparison to QPE, outlining regimes where DB-QITE offers advantages in near-term to early fault-tolerant quantum computing for ground-state preparation.

Abstract

Efficiently preparing approximate ground-states of large, strongly correlated systems on quantum hardware is challenging and yet nature is innately adept at this. This has motivated the study of thermodynamically inspired approaches to ground-state preparation that aim to replicate cooling processes via imaginary-time evolution. However, synthesizing quantum circuits that efficiently implement imaginary-time evolution is itself difficult, with prior proposals generally adopting heuristic variational approaches or using deep block encodings. Here, we use the insight that quantum imaginary-time evolution is a solution of Brockett's double-bracket flow and synthesize circuits that implement double-bracket flows coherently on the quantum computer. We prove that our Double-Bracket Quantum Imaginary-Time Evolution (DB-QITE) algorithm inherits the cooling guarantees of imaginary-time evolution. Concretely, each step is guaranteed to i) decrease the energy of an initial approximate ground-state by an amount proportion to the energy fluctuations of the initial state and ii) increase the fidelity with the ground-state. We provide gate counts for DB-QITE through numerical simulations in Qrisp which demonstrate scenarios where DB-QITE outperforms quantum phase estimation. Thus DB-QITE provides a means to systematically improve the approximation of a ground-state using shallow circuits.

Figures (6)

• Figure 1: Double-bracket Quantum Imaginary-Time Evolution (DB-QITE). We propose a new quantum algorithm to implement imaginary-time evolution (ITE). To implement the Quantum Imaginary-Time Evolution (QITE) unitary $Q_{\tau}$, we utilize a Double-Bracket Quantum Algorithm (DBQA) and show that QITE can be recursively compiled using Hamiltonian evolution and reflection gates.
• Figure 2: Numerical benchmarks for the 1d Heisenberg model. (a) DB-QITE for $L=20$ qubits starting with either a product of singlets $|\text{Singlet}\rangle$ or a low-depth variationally learnt 'warm-start' state $|\text{HVA}\rangle$ based on Refs. bosse_Heisenberg_2022kattemolle_vanwezel_2022wecker2015progress. When extending $s_k$, just a handful of steps allows to reach $E_k\ll \lambda_1$. (b) Two DB-QITE steps yield fidelities $F_2\ge 95\%$ for $L=12$ and $F_2\ge 90\%$ for $L=20$ by using approximately $2\times 10^3$ CZ gates. QPE (orange) outperforms DB-QITE (blue) only when fault-tolerant execution allows for circuits with more than $10^4$ CZ gates. In contrast, DB-QITE achieves high fidelity using fewer gates and more homogeneous circuits (See App. \ref{['app Qrisp']} for details). (c) QPE ideally uses a rescaled Hamiltonian $\hat{H}'= (\hat{H}-\lambda_0\mathmybb{1})/(\|\hat{H}\|-\lambda_0)$, but overestimating $|| \hat{H}||$ (e.g., by a factor of 2) significantly increases QPE runtime. In such cases, DB-QITE can attain fidelity $F_4\approx 95\%$ more efficiently.
• Figure 3: 5 iterations of DB-QITE for $L$-qubit Heisenberg model. The figures show the energies $E_k=\langle \omega_k | \hat{H} | \omega_k \rangle$ and the fidelities $F_k=|\langle \omega_k | \lambda_0 \rangle|^2$ with the ground state $|\lambda_0\rangle$ for the $k$-th iterates $|\omega_k\rangle$ for initializations $|\omega_0\rangle=|\text{Singlet}\rangle$ and $|\omega_0\rangle=|\text{HVA}\rangle$. The upper (dashed, orange) lines indicate the fidelities $1.0, 0.975, 0.95$, and the lower (dashed, red) lines indicate the ground state energies $\lambda_0$ and the first excited state energies $\lambda_1$. Let us make the following observations: 1) The $|\text{HVA}\rangle$ initializations achieve a higher fidelity to the ground state compared to the $|\text{Singlet}\rangle$ initializations, and the fidelity of the $k$-th iterates for $|\text{HVA}\rangle$ consistently exceeds the respective fidelities for $|\text{Singlet}\rangle$. However, this gap narrows significantly for an increasing number of DB-QITE steps $k$. Therefore, the choice of the particular initial state can be less impactful as long as there is some overlap with the ground state. 2) The optimal evolution times $s_k$ found iteratively by a 20-point grid search tend to decrease with $k$, while remaining orders of magnitude larger than guaranteed by Theorem \ref{['th: fidelity convergence']}. In particular, large initial evolution times $s_0$ yield a rapid decrease in energy in the first step of DB-QITE. Similarly, we observe that the reached fidelities $F_k$ are much larger than guaranteed by Theorem \ref{['th: fidelity convergence']} by optimizing the $s_k$ durations xiaoyue2024strategies. For example, for $L=10$ we empirically observe $F_k>1-q^k$ for $q=0.5$, while Theorem \ref{['th: fidelity convergence']} would merely provide a theoretical lower bound with $q$ being close to 1. 3) The fidelity $F_k$ with the ground state $|\lambda_0\rangle$ reached for the $k$-th iterate decreases with increasing number of qubits $L$, e.g., for 5 steps of DB-QITE with $|\text{HVA}\rangle$ initialization, fidelities $F_5=0.999$ for $L=10$, and $F_5=0.967$ for $L=20$ are achieved.
• Figure 4: Gate counts, circuit depth and ground-state fidelity for DB-QITE (left column) and QPE (right column) for the Heisenberg model with $L=12,16$ and $20$ qubits for $|\text{HVA}\rangle$ and $|\text{Singlet}\rangle$ (dashed) as initializations. For $L=12$ the ground state is prepared with fidelity $F_3 = 99\%$ with $k=3$ steps of DB-QITE. The circuit implementing this requires $N_\text{CZ}\approx 5.5 \times 10^3$ CZ gates and $N_\text{U3}\approx 8.8 \times 10^3$ single-qubit gates. For $L=20$ the ground state is prepared with fidelity $F_2 = 92\%$ with $k=2$ steps of DB-QITE. The circuit implementing this requires $N_\text{CZ}\approx 3 \times 10^3$ CZ gates and $N_\text{U3}\approx 4.8 \times 10^3$ single-qubit gates. The structure of DB-QITE's circuits is more homogeneous and e.g. for $k=4$ DB-QITE has circuit depth $D_4 \approx 1.2\times 10^4$ reaching $F_4 \approx 96\%$ which is lower circuit depth of QPE with $k=4$ precision qubits given by $D_4\approx 1.3 \times 10^4$ giving only $F_4 \approx 95\%$. When considering long enough circuits then QPE reaches enough precision ($k$ for QPE denotes the number of auxiliary qubits encoding precision) and under the assumption of all-to-all connectivity outperforms DB-QITE by reaching higher fidelity through less gates and lower circuit depth.
• Figure 5: Comparison between DB-QITE and QPE for $L$-qubit Heisenberg model for $|\text{Singlet}\rangle$ (top) and $|\text{HVA}\rangle$ (bottom) initializations. Fidelities $F_k$ with the ground state $|\lambda_0\rangle$ and number of CZ gate are shown as a function of steps $k=0,1,\dotsc,5$ for DB-QITE (left) and for QPE (right) as a function of the number of precision qubits $k=1,2,\dotsc,5$. The dashed-orange lines indicate the fidelities $1.0, 0.975, 0.95$. For example, for 10 qubits just $k=4$ steps of DB-QITE are required to reach almost perfect fidelity, which is achieved with QPE with $k=3$ precision qubits. Both methods require exponential gate counts but QPE converges faster than DB-QITE. For $|\text{HVA}\rangle$ initializations and $L=20$ qubits, a fidelity $F\approx 95\%$ is reached with $k=3$ steps of DB-QITE, or QPE with $k=4$ precision qubits, requiring $N_{\text{CZ}}\approx 10^4$ CZ gates in both cases. While DB-QITE deterministically prepares the (approximate) ground state, QPE has a success probability of $P\approx 79\%$. If QPE suffers from an inaccurate spectrum rescaling to $[0,0.5)$, the fidelity for $k=4$ precision qubits drops to $F\approx 86\%$ and an additional precision qubit is required. In this scenario DB-QITE clearly outperforms QPE.

Theorems & Definitions (27)

• Theorem 1
• Theorem 2
• Proposition 4: ITE is a solution to a DBF
• proof
• Remark
• Proposition 5: ITE Fluctuation-refrigeration relation
• Proposition 5: ITE Fluctuation-refrigeration relation
• proof
• Remark
• Proposition 11: QITE-DBI Fluctuation-refrigeration relation