Energy shifting is usually trivial in imaginary-time evolution because it changes only the normalization of the evolved state. On a quantum computer, however, imaginary-time evolution can be implemented as a coherent or sampled superposition of real-time evolutions, in which energy shifts generate relative phases that can interfere. Here we introduce multi-shift quantum imaginary-time evolution (MS-QITE), which uses a distribution of energy shifts to engineer interference and optimize different implementations. In a Monte Carlo realization, multi-shift reshapes the sampling distribution and concentrates it within a shorter real-time window, thereby reducing the required Hamiltonian-evolution time and improving the stability of ground-state-energy estimation. In a continuous-variable-assisted realization, it permits postselection over a finite quadrature interval rather than near a single quadrature value, substantially reducing resource consumption while retaining accurate thermal-state preparation. Numerical results for transverse-field Ising models demonstrate that energy shifts provide a useful interference-based approach for optimizing quantum imaginary-time evolution.
Energy shifting is usually trivial in imaginary-time evolution because it changes only the normalization of the evolved state. On a quantum computer, however, imaginary-time evolution can be implemented as a coherent or sampled superposition of real-time evolutions, in which energy shifts generate relative phases that can interfere. Here we introduce multi-shift quantum imaginary-time evolution (MS-QITE), which uses a distribution of energy shifts to engineer interference and optimize different implementations. In a Monte Carlo realization, multi-shift reshapes the sampling distribution and concentrates it within a shorter real-time window, thereby reducing the required Hamiltonian-evolution time and improving the stability of ground-state-energy estimation. In a continuous-variable-assisted realization, it permits postselection over a finite quadrature interval rather than near a single quadrature value, substantially reducing resource consumption while retaining accurate thermal-state preparation. Numerical results for transverse-field Ising models demonstrate that energy shifts provide a useful interference-based approach for optimizing quantum imaginary-time evolution.
Abstract We introduce the framework of model space into quantum imaginary time evolution (QITE) to enable stable estimation of ground and excited states using a quantum computer. Model-space QITE (MSQITE) propagates a model space to the exact one by retaining its orthogonality, and hence is able to describe multiple states simultaneously. The quantum Lanczos (QLanczos) algorithm is extended to MSQITE to accelerate the convergence. The present scheme is found to outperform both the standard QLanczos and the recently proposed folded-spectrum QITE in simulating excited states. Moreover, we demonstrate that spin contamination can be effectively removed by shifting the imaginary time propagator, and thus excited states with a particular spin quantum number are efficiently captured without falling into the different spin states that have lower energies. We also investigate how different levels of the unitary approximation employed in MSQITE can affect the results. The effectiveness of the algorithm over QITE is demonstrated by noise simulations for the H4model system.
Quantum Imaginary-Time Evolution (QITE) is a powerful method for preparing ground states on quantum hardware. However, executing QITE has costly measurement budgets for general Hamiltonians. Both fidelity and computational cost are strongly dependent on the definition of suitable local domains and Hamiltonian partitions. In this work, we introduce the Multiple-Time QITE algorithm (MT-QITE). We show how using more than one imaginary time substantially improves the fidelity of the resulting ground state as well as the measurement overhead with respect to the previously published QITE algorithm, while preserving its deterministic character and its independence from ad hoc ansatze. Moreover, unlike QITE and other QITE-based algorithms, MT-QITE is parallelizable, and we show that even in Hamiltonians with non-local interactions, partitioning may entail a computational advantage.
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.
Imaginary-time evolution (ITE) is a foundational method for ground state preparation on quantum computers. However, because ITE is non-unitary, existing implementations incur a sample complexity and/or classical cost that scales exponentially with the target imaginary time β. Moreover, the state itself converges slower than the energy, making accurate estimation of arbitrary ground state observables even more expensive. In this work, we improve upon standard ITE by introducing twirled imaginary-time evolution (TITE), which pairs ITE with real-time evolution applied for a random duration drawn from a carefully designed distribution. We prove that this randomization quadratically suppresses the trace distance to the ground state, and thus also the error of arbitrary observables, which allows roughly half of the imaginary time to be replaced with real-time evolution (β↦β/2) while maintaining the same level of accuracy. Because real-time evolution is unitary and does not incur an overhead in sample complexity or classical computation, this affords a quadratic reduction in the cost of any black-box ITE implementation, including Trotterization and quantum imaginary-time evolution. We demonstrate the efficiency of our algorithm in noisy circuit-level simulations of a non-integrable Ising chain, showing substantial improvements over standard ITE.
Abstract Most quantum algorithms designed to generate or probe properties of the ground state of a quantum many-body system require as input an initial state with a large overlap with the desired ground state. One approach for preparing such a ground state is Imaginary Time Evolution (ITE). Recent work by [Motta, M. , Sun, C. , Tan, A.T.K. et al (2020)] introduced an algorithm—which we will refer to as Quantum Imaginary Time Evolution (QITE)—that shows how ITE can be approximated by a sequence of unitary operators, making QITE potentially implementable on early fault-tolerant quantum computers. In this work, we provide a heuristic study of the capabilities of the QITE algorithm in approximating the ITE of lattice and molecular electronic structure Hamiltonians. We numerically study the performance of the QITE algorithm when provided with a good classical initial state for a large class of systems, some of which are of interest to industrial applications, and check if QITE is able to qualitatively replicate the ITE behavior and improve over a classical mean-field solution. The systems we consider in this work range from one- and two-dimensional lattice systems of various lattice geometries displaying short- and long-range interactions, to active spaces of molecular electronic structure Hamiltonians. In addition to the comparison of QITE and ITE, we explicitly show how imaginary time evolved fermionic Gaussian states can serve as initial states which can be efficiently computed on classical computers and efficiently implemented on quantum computers for generic spin Hamiltonians in arbitrary lattice geometries and dimensions, which can be of independent interest.
Quantum imaginary-time evolution (QITE) is a promising tool to prepare thermal or ground states of Hamiltonians, as convergence is guaranteed when the evolved state overlaps with the ground state. However, its implementation using a a hybrid quantum-classical approach, where the dynamics of the parameters of the quantum circuit are derived by McLachlan's variational principle, is impractical as the number of parameters m increases, since each step in the evolution takes Θ ( m 2 ) state preparations to calculate the quantum Fisher information matrix (QFIM). In this work, we accelerate QITE by rapid estimation of the QFIM, while conserving the convergence guarantees to the extent possible. To this end, we prove that if a parameterized state is rotated by a 2-design and measured in the computational basis, then the QFIM can be inferred from partial derivative cross correlations of the probability outcomes. One sample estimate costs only Θ ( m ) state preparations, leading to rapid QFIM estimation when a few samples suffice. The second family of estimators takes greater liberties and replace QFIMs with averaged classical Fisher information matrices (CFIMs). In an extreme special case optimized for rapid (over accurate) descent, just one CFIM sample is drawn. We justify the second estimator family by proving rapid descent. Guided by these results, we propose the algorithm, which we showcase and test in several molecular systems, with the goal of preparing ground states.