Vari-Cool: a non-unitary quantum variational protocol for simulated cooling

TL;DR

Vari-Cool presents a non-unitary variational framework that drives a quantum system toward low-energy states of a target Hamiltonian by repeating cycles of a parameterized unitary block on system-plus-bath qubits followed by bath resets. The method optimizes a shallow, tunable circuit to minimize the steady-state energy $E_{\rm steady}=\mathrm{Tr}[\rho_{\rm steady}\hat{H}_{\rm sys}]$, enabling rapid cooling within a small number of cycles. Demonstrations on the TFIM show transferability from small classical training systems ($N=4$) to larger ones ($N\le 28$) and robustness to noise, with experimental validation on IBM's ibm_kingston achieving substantial portion of the ground-state energy. The work highlights the potential of dissipative, variational approaches for NISQ devices, especially where nonlocal excitations impede purely unitary ground-state preparation, and outlines future directions toward fermionic baths and Gibbs-state preparation.

Abstract

We introduce a variational approach for preparing low energy states of arbitrary target Hamiltonians. The protocol is defined in terms of a repeated cycle consisting of p layers of unitary gates applied to the system and ancilla "bath" qubits, followed by reset of the bath qubits. The gate parameters within each cycle are optimized such that the steady state achieved after many cycles has a low energy expectation value with respect to the target Hamiltonian, and that the energy converges toward the steady state value in as few cycles as possible. We illustrate the protocol for the transverse field Ising model, and study its systematic behaviors with respect to system size, model parameters, and noise using tensor network based classical simulations. We then experimentally demonstrate its operation on IBM's ibm_kingston quantum processor for up to 28 system qubits coupled to 14 bath sites. Classical training on small system sizes and with few unitary layers per cycle gives robust results that transfer well to larger system sizes and to noisy hardware.

Figures (14)

• Figure 1: The Vari-Cool state preparation protocol, as applied to the transverse field Ising model. a) Setup and labeling for the case of $N$ system qubits coupled to $n_{\rm bath} = N/2$ bath qubits. The choice $n_{\rm bath} = N/2$ is convenient for the experimental setup in Sec. \ref{['sec:QPU']}, but is not fundamental to the protocol. b) Low energy states are prepared by repeatedly applying a non-unitary cycle composed of unitary gates acting on system and ancilla "bath" qubits, and bath qubit reset operations. Within each cycle, $p$ layers of unitary gates are applied as shown, with rotation angles $\alpha_\ell$, $\beta_\ell$, $\gamma_\ell$, and $\delta_\ell$ chosen to produce a steady state (achieved after many cycles) with low energy evaluated with respect to the system Hamiltonian $\hat{H}_{\rm sys}$.
• Figure 2: Example of training the cooling circuit. Parameters are optimized on a classical noiseless simulator with $N = 4$ system qubits and $n_{\rm bath} = 2$ bath qubits, and $p = 3$ layers in the unitary block. Here we show the case $J = 0.4$, $h = 0.6$. a) The black solid line shows the residual energy density $(E - E_0)/N$ after $T_{\rm train} = 7$ cycles; this value is used for the optimization in each training iteration. Here $E_0 = -2.6016$ is the ground state energy (in the units above where $J + h = 1$). The gray line (below the black line) shows the steady state energy density achieved by applying the circuit 40 times with the current values of the parameters at each training iteration. See Sec. \ref{['sec:training']} for details of the training. b) Residual energy density as a function of the number of cycles applied, obtained with the final optimized parameters from the training in panel a. The system qubits were initialized in the state $|0000\rangle$. The residual energy density after $40$ cycles is $(E - E_0)/N = 7.7 \times 10^{-3}$ per site.
• Figure 3: Dependence of the steady state energy density on two qubit gate error probability, from stochastic evolution of matrix product states. Different system sizes $N=4,8,16,28$ (with $n_{\rm{bath}}=N/2$) are represented by different line styles. Left panel: Energy density relative to the ground state, $(E_{\rm steady}-E_0)/N$, in the paramagnetic phase with $J = 0.4$, $h = 0.6$. Right panel: Energy density relative to the ground state in the ferromagnetic phase with $J = 0.6$, $h = 0.4$. The larger overall scale and system size dependence of the energy density in the ferromagnetic phase reflect the challenge of cooling topological domain wall excitations in the ferromagnetic phase mathhies2022adibatic_demag.
• Figure 4: Steady state spin-spin correlations, from classical simulations. a) Spin-spin correlation function, at four representative points in the phase diagram at the steady state reached in the absence of noise ("st.," dark lines) and in the ground state ("gr.," light lines). The system reaches a broken symmetry state with finite magnetization in the ferromagnetic phase, resulting in the plateau seen for $J = 0.6, h = 0.4$. b) Spin-correlation function for various noise levels at $J = 0.6, h = 0.4$. The long-ranged ferromagnetic correlations are rapidly washed out by the presence of noise.
• Figure 5: Qubit layout for experimental runs on the ibm_kingston quantum processor. To minimize waiting times and potential cross-talk errors during bath qubit RESET operations on the quantum processor, we use two qubits for each bath site. At the RESET step of the protocol in Fig. \ref{['fig:setup']}b, we apply a SWAP operation between the proximal and distal bath qubits (see labeling on figure), then apply a RESET to the distal bath qubit in parallel with the next unitary block.