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Imaginary-time evolution algorithms for intermediate-scale quantum signal processors

Thais de Lima Silva, Márcio M. Taddei, Stefano Carrazza, Leandro Aolita

Published Oct 25, 2021• 0 citations
OpenAlex

Abstract

Simulating quantum imaginary-time evolution (QITE) with high precision is a major promise of quantum computation. However, the known algorithms are either probabilistic (repeat until success) with unpractically small success probabilities or coherent (quantum amplitude amplification) but with circuit depths and ancillary-qubit numbers unrealistically large for the mid-term. We introduce a new type of high-precision algorithms amenable to mid-term devices. First, we present two QITE primitives featuring excellent complexity scalings. In fact, one of them is optimal in ancillary overhead (requiring a single ancillary qubit throughout) whereas the other one is optimal in runtime for small inverse temperature $\beta$ or high precision $\varepsilon^{-1}$. The latter is shown by noting that the runtime saturates a cooling-speed limit that is the imaginary-time counterpart of the no fast-forwarding theorem of real-time simulations, which we prove. Our primitives are based on the quantum signal processing formalism for operator-function synthesis, to which we make two technical contributions relevant beyond QITE. Second, we present a master algorithm for deterministic QITE with an overall runtime asymptotically better than that of coherent approaches and the same hardware requirements as probabilistic ones, remarkably. This is based on a surprisingly simple idea: partitioning the evolution into several low-$\beta$ fragments that are successively run probabilistically. Our results are relevant to near-term quantum hardware.

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Simulating quantum imaginary-time evolution (QITE) with high precision is a major promise of quantum computation. However, the known algorithms are either probabilistic (repeat until success) with unpractically small success probabilities or coherent (quantum amplitude amplification) but with circuit depths and ancillary-qubit numbers unrealistically large for the mid-term. We introduce a new type of high-precision algorithms amenable to mid-term devices. First, we present two QITE primitives featuring excellent complexity scalings. In fact, one of them is optimal in ancillary overhead (requiring a single ancillary qubit throughout) whereas the other one is optimal in runtime for small inverse temperature β\betaβ or high precision ε−1\varepsilon^{-1}ε−1. The latter is shown by noting that the runtime saturates a cooling-speed limit that is the imaginary-time counterpart of the no fast-forwarding theorem of real-time simulations, which we prove. Our primitives are based on the quantum signal processing formalism for operator-function synthesis, to which we make two technical contributions relevant beyond QITE. Second, we present a master algorithm for deterministic QITE with an overall runtime asymptotically better than that of coherent approaches and the same hardware requirements as probabilistic ones, remarkably. This is based on a surprisingly simple idea: partitioning the evolution into several low-β\betaβ fragments that are successively run probabilistically. Our results are relevant to near-term quantum hardware.
Simulating quantum imaginary-time evolution (QITE) is a significant promise of quantum computation. However, the known algorithms are either probabilistic (repeat until success) with unpractically small success probabilities or coherent (quantum amplitude amplification) with circuit depths and ancillary-qubit numbers unrealistically large in the mid-term. Our main contribution is a new generation of deterministic, high-precision QITE algorithms that are significantly more amenable experimentally. A surprisingly simple idea is behind them: partitioning the evolution into a sequence of fragments that are run probabilistically. It causes a considerable reduction in wasted circuit depth every time a run fails. Remarkably, the resulting overall runtime is asymptotically better than in coherent approaches, and the hardware requirements are even milder than in probabilistic ones. Our findings are especially relevant for the early fault-tolerance stages of quantum hardware.
Quantum imaginary time evolution (QITE) is one of the promising candidates for finding the eigenvalues and eigenstates of a Hamiltonian on a quantum computer. However, the original proposal suffers from large circuit depth and measurements due to the size of the Pauli operator pool and Trotterization. To alleviate the requirement for deep circuits, we propose a time-dependent drifting scheme inspired by the qDRIFT algorithm [Campbell, E. Phys. Rev. Lett. 2019, 123, 070503]. We show that this drifting scheme removes the depth dependency on the size of the operator pool and converges inversely with respect to the number of steps. We further propose a deterministic algorithm that selects the dominant Pauli term to reduce the fluctuation for the ground state preparation. We also introduce an efficient measurement reduction scheme across Trotter steps that removes its cost dependence on the number of iterations. We analyze the main source of error for our scheme both theoretically and numerically. We numerically test the validity of depth reduction, convergence performance of our algorithms, and the faithfulness of the approximation for our measurement reduction scheme on several benchmark molecules. In particular, the results on the LiH molecule give circuit depths comparable to that of the advanced adaptive variational quantum eigensolver (VQE) methods while requiring much fewer measurements.
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.
A probabilistic imaginary-time evolution (PITE) method was proposed as a nonvariational method to obtain a ground state on a quantum computer. In this formalism, the success probability of obtaining all imaginary-time evolution operators acting on the initial state decreases as the imaginary time proceeds. To alleviate the undesirable nature, we propose quantum circuits for PITE combined with the quantum amplitude amplification (QAA) method. We reduce the circuit depth in the combined circuit with QAA by introducing a pre-amplification operator. We successfully demonstrated that the combination of PITE and QAA works efficiently and reported a case in which the quantum acceleration is achieved. Additionally, we have found that by optimizing a parameter of PITE, we can reduce the number of QAA operations and that deterministic imaginary-time evolution (deterministic ITE) can be achieved which avoids the probabilistic nature of PITE. We applied the deterministic ITE procedure to multiple imaginary-time steps and discussed the computational cost for the circuits. Finally, as an example, we demonstrate the numerical results of the PITE circuit combined with QAA in the first- and second-quantized Hamiltonians.
Abstract Imaginary-time evolution is fundamental for analyzing quantum many-body systems, with applications spanning quantum chemistry, condensed matter physics, and quantum field theory, yet classical simulation requires exponentially growing resources in both system size and evolution time. While quantum approaches reduce the system-size scaling, existing methods rely on heuristic techniques with measurement precision or success probability that deteriorates as evolution time increases. We present a quantum algorithm that prepares normalized imaginary-time evolved states using an adaptive normalization factor to maintain a stable success probability over long imaginary-time intervals. Our algorithm approximates the target state with error polynomially small in the inverse imaginary time using a polynomial number of elementary quantum gates and a single ancilla qubit, with success probability close to one. When the initial state has reasonable overlap with the ground state, this algorithm also achieves polynomial query complexity in the system size. To our knowledge, this is the first quantum algorithm for imaginary-time evolution with provably polynomial resource scaling in evolution time. Numerical experiments validate our theoretical analysis for evolution time up to 50, demonstrating the algorithm’s effectiveness for long-time evolution. Building on this technique, we further develop imaginary-time-evolution-based algorithms for ground-state-related problems and for simulating open quantum systems. These algorithms can reduce circuit depth in certain regimes compared with existing methods, at the expense of higher total query complexity, advancing the practical feasibility of quantum simulation on early fault-tolerant devices.
Dynamic quantum circuits combine mid-circuit measurement with classical feed-forward, enabling circuit constructions with reduced entangling-gate depth. Here, we investigate their use in Quantum Imaginary Time Evolution (QITE), where circuit depth and parameter growth limit practical implementations of ground-state preparation. For dense classical optimization Hamiltonians, we introduce a reduced-parameter QITE ansatz that restricts entanglement generation via a small set of control qubits, enabling each QITE layer to be implemented with constant two-qubit gate depth using fan-out-based dynamic circuits. In noiseless simulations of exact cover and set partitioning instances, the reduced ansatz yields a higher success probability than standard QITE approaches. We implement unitary, dynamic fan-out, and semi-classical adaptive variants on IBM superconducting hardware. The semi-classical variant performs favorably to the unitary implementation, while the fully dynamic construction exposes the trade-offs between entangling-depth reduction and measurement and feed-forward overhead associated to dynamic circuit implementations. Using a fidelity threshold of 0.5 relative to the noiseless QITE ansatz, we show that dynamic fan-out based QITE would outperform unitary implementations on current devices when the measurement and two-qubit gate errors are reduced by 65% and the feedback latency is halved.