Double-bracket quantum algorithms for high-fidelity ground-state preparation
Matteo Robbiati, Edoardo Pedicillo, Andrea Pasquale, Li Xiaoyue, Oriel Kiss, Andrew L. Wright, Renato M. S. Farias, Khanh Uyen Giang, Jeongrak Son, Johannes Knörzer, Siong Thye Goh, Jun Yong Khoo, Nelly H. Y. Ng, Zoë Holmes, Stefano Carrazza, Marek Gluza
Ground-state preparation is a central application for quantum computers but remains challenging in practice. In this work, we quantitatively investigate the performance and gate counts of double-bracket quantum algorithms (DBQAs) for ground-state preparation. We propose a practical strategy in which DBQAs refine initial state preparation circuits, and we compile them for Heisenberg chains using controlled-Z and single-qubit gates. Warm-started DBQAs consistently improve both the energy and ground-state fidelity relative to the initial states provided by variational ansätze, indicating that DBQAs offer an effective unitary synthesis method. To demonstrate compatibility with near-term hardware, we executed a proof-of-concept example on IBM devices. With error mitigation, we observed a statistically significant improvement over the corresponding warm-start circuit. Furthermore, numerical emulations for the same system size indicate that executing DBQAs on Quantinuum's hardware could achieve similar cost-function gains without requiring error mitigation. These findings suggest that DBQAs are a promising approach for enhancing ground-state approximations on near-term quantum devices.
Ground-state preparation is a central application for quantum computers but remains challenging in practice. In this work, we quantitatively investigate the performance and gate counts of double-bracket quantum algorithms (DBQAs) for ground-state preparation. We propose a practical strategy in which DBQAs refine initial state preparation circuits, and we compile them for Heisenberg chains using controlled-Z and single-qubit gates. Warm-started DBQAs consistently improve both the energy and ground-state fidelity relative to the initial states provided by variational ansätze, indicating that DBQAs offer an effective unitary synthesis method. To demonstrate compatibility with near-term hardware, we executed a proof-of-concept example on IBM devices. With error mitigation, we observed a statistically significant improvement over the corresponding warm-start circuit. Furthermore, numerical emulations for the same system size indicate that executing DBQAs on Quantinuum's hardware could achieve similar cost-function gains without requiring error mitigation. These findings suggest that DBQAs are a promising approach for enhancing ground-state approximations on near-term quantum devices.
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 proportional 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 that 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.
We propose quantum algorithms for preparing thermal states via the simulation of the thermofield double states. The key idea is to leverage double-bracket quantum algorithms to implement imaginary-time evolution on thermofield double states, whose reduced state realizes the Gibbs state. Our method, termed double-bracket thermofield double (DB-TFD), introduces two variants. The first, the vanilla DB-TFD algorithm, directly implements imaginary-time evolution using double-bracket quantum imaginary-time evolution. The second, poly DB-TFD, employs double-bracket quantum signal processing to approximate the imaginary-time evolution operator via a polynomial transformation. We demonstrate that the complexity of the poly DB-TFD algorithm scales exponentially with the inverse temperature in a broad practical regime. This scaling is consistent with existing methods, and numerical simulations support the corresponding theoretical bound. We further demonstrate the utility of DB-TFD in quantum Boltzmann machines for generative modeling, achieving improved performance compared with variational imaginary-time evolution approaches. These results establish DB-TFD as a promising route for thermal state preparation in the near-term and early-fault-tolerant regimes.
Quantum Signal Processing (QSP), a framework for implementing matrix-valued polynomials, is a fundamental primitive in various quantum algorithms. Despite its versatility, a potentially underappreciated challenge is that all systematic protocols for implementing QSP rely on post-selection. This can impose prohibitive costs for tasks when amplitude amplification cannot sufficiently improve the success probability. For example, in the context of ground-state preparation, this occurs when using a too poor initial state. In this work, we introduce a new formula for implementing QSP transformations of Hermitian matrices, which requires neither auxiliary qubits nor post-selection. Rather, using approximation to the exact unitary synthesis, we leverage the theory of the double-bracket quantum algorithms to provide a new quantum algorithm for QSP, termed Double-Bracket QSP (DB-QSP). The algorithm requires the energy and energetic variance of the state to be measured at each step and has a recursive structure, which leads to circuit depths that can grow super exponentially with the degree of the polynomial. With these strengths and caveats in mind, DB-QSP should be viewed as complementing the established QSP toolkit. In particular, DB-QSP can deterministically implement low-degree polynomials to ``warm start" QSP methods involving post-selection.
This work proposes double-bracket iterations as a framework for obtaining diagonalizing quantum circuits. Their implementation on a quantum computer consists of interlacing evolutions generated by the input Hamiltonian with diagonal evolutions which can be chosen variationally. No qubit overheads or controlled-unitary operations are needed but the method is recursive which makes the circuit depth grow exponentially with the number of recursion steps. To make near-term implementations viable, the proposal includes optimization of diagonal evolution generators and of recursion step durations. Indeed, thanks to this numerical examples show that the expressive power of double-bracket iterations suffices to approximate eigenstates of relevant quantum models with few recursion steps. Compared to brute-force optimization of unstructured circuits double-bracket iterations do not suffer from the same trainability limitations. Moreover, with an implementation cost lower than required for quantum phase estimation they are more suitable for near-term quantum computing experiments. More broadly, this work opens a pathway for constructing purposeful quantum algorithms based on so-called double-bracket flows also for tasks different from diagonalization and thus enlarges the quantum computing toolkit geared towards practical physics problems.
Quantum mechanical ground states of many-body systems can be important resources for various investigations: for quantum sensing, for benchmarking quantum hardware with classically solvable states, as the initial states for nonequilibrium quantum dynamics following quenches, the simulation of quantum processes that start by coupling systems in ground states, eg, could be a process in quantum chemistry, while their approximations are required as inputs to quantum phase estimation algorithm. However, preparing ground states can be challenging; for example, it may require adiabatic switching of Hamiltonian terms slower than an inverse gap, which can be time consuming and bring in decoherence. Here we investigate the possibility of preparing a many-body entangled ground state of a certain Hamiltonian, which can be called a quantum ``problem'' Hamiltonian, using the time evolution of an initial fiducial state by another time independent ``solver'' Hamiltonian with couplings up to unit strength for a very short fixed (unit) time: a ``time efficient'' ansatz. The parameters of the solver Hamiltonian are optimised classically minimising energy as the cost function. We present a study of up to n=14 qubit many-body states prepared using this methodology. Importantly, we find that a strategy of combining a warm start (an already prepared ground state of a n−1 qubit Hamiltonian) and incrementally adding extra couplings of a qubit is the best scaling strategy to prepare the ground state of a n-qubit Hamiltonian.
Imaginary-time evolution (ITE) is a widely used technique for preparing the ground state of a target Hamiltonian. Building on the recently proposed framework of Double-Bracket quantum ITE (DB-QITE), we introduce a probabilistic variant (PDBQITE) that achieves an exponential reduction in circuit depth. Our algorithm requires only a single ancilla qubit and a quantum circuit that consists of one controlled real-time evolution and one mid-circuit measurement per iteration. The trade-off is an exponential decay of the total success probability with the number of iterations, for which we derive an analytical lower bound that is independent of system size. Our numerical simulations show that the per-step success probability remains close to unity, making it realistic to execute many iterations, thus providing a substantial improvement over prior constructions. We show that PDBQITE outperforms the near-term probabilistic algorithm PITE for ground-state preparation of molecular Hamiltonians, and that a hybrid QAOA-PDBQITE scheme achieves higher approximation ratios than QAOA at the same circuit depth on random regular graphs for the MaxCut problem.