We introduce a method to perform imaginary time evolution in a controllable quantum system using measurements and conditional unitary operations. By performing a sequence of weak measurements based on the desired Hamiltonian constructed by a Suzuki-Trotter decomposition, an evolution approximating imaginary time evolution can be realized. The randomness due to measurement is corrected using conditional unitary operations, making the evolution deterministic. Both the measurements required for the algorithm and the conditional unitary operations can be constructed efficiently. We show that the algorithm converges only below a specified energy threshold and the complexity is estimated for some specific problem instances.
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 combinatio…
Imaginary time evolution is a powerful tool for studying quantum systems. While it is possible to simulate with a classical computer, the time and memory requirements generally scale exponentially with the system size. Conversely, quantum computers can efficiently simulate quantum systems, but not non-unitary imaginary time evolution. We propose a variational algorithm for simulating imaginary time evolution on a hybrid quantum computer. We use this algorithm to find the ground-state energy of many-particle systems; specifically molecular hydrogen and lithium hydride, finding the ground state …
Ground state preparation in $(2+1)$-dimensional pure $\mathbb{Z}_2$ lattice gauge theory via deterministic quantum imaginary time evolution
The University of Tokyo（东京大学）；Keio University（庆应义塾大学）；International Center for Elementary Particle Physics (ICEPP), The University of Tokyo（东京大学基本粒子物理国际中心）
Simulation of quantum matters is a significant application of quantum computers. In contrast to the unitary operation which can be realized naturally on a quantum computer, the implementation of nonunitary operation, widely used in classical approaches, needs special designing. Here, by application of Grover's algorithm, we extend the probabilistic method of implementing nonunitary operation and show that it can promote success probability without fidelity decreasing. This method can be applied to problems of imaginary time evolution and contraction of tensor networks on a quantum computer.
Imaginary time evolution is a powerful tool applied in quantum physics, while existing classical algorithms for simulating imaginary time evolution suffer high computational complexity as the quantum systems become larger and more complex. In this work, we propose a probabilistic algorithm for implementing imaginary time evolution based on non-unitary quantum circuit. We demonstrate the feasibility of this method by solving the ground state energy of several quantum many-body systems, including H2, LiH molecules and the quantum Ising chain. Moreover, we perform experiments on superconducting a…
Double-bracket quantum algorithms for quantum imaginary-time evolution
Nanyang Technological University（南洋理工大学）；Fraunhofer Institute FOKUS（弗劳恩霍夫FOKUS研究所）；Ecole Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院）；Keio University（庆应义塾大学）；National University of Singapore（新加坡国立大学）
The Gibbs partition function is an important quantity in describing statistical properties of a system in thermodynamic equilibrium. There are several proposals to calculate the partition functions on near-team quantum computers. However, the existing schemes require many copies of the Gibbs states to perform an extrapolation for the calculation of the partition function, and these could be costly performed on the near-term quantum computers. Here, we propose an efficient scheme to calculate the Gibbs function with the imaginary time evolution. To calculate the Gibbs function of $N$ qubits, on…
The quantum imaginary time evolution is a powerful algorithm for preparing the ground and thermal states on near-term quantum devices. However, algorithmic errors induced by Trotterization and local approximation severely hinder its performance. Here we propose a deep reinforcement learning-based method to steer the evolution and mitigate these errors. In our scheme, the well-trained agent can find the subtle evolution path where most algorithmic errors cancel out, enhancing the fidelity significantly. We verified the method's validity with the transverse-field Ising model and the Sherrington-…
Interference Engineering for Quantum Imaginary-Time Evolution through Multiple Energy Shifts
Key Laboratory of Atomic and Subatomic Structure and Quantum Control (Ministry of Education), Guangdong Basic Research Center of Excellence for Structure and Fundamental Interactions of Matter, and School of Physics, South China Normal University（华南师范大学物理学院，原子和亚原子结构及量子控制教育部重点实验室，广东省物质结构和基本相互作用卓越基础研究中心）；Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, Guangdong-Hong Kong Joint Laboratory of Quantum Matter, and Frontier Research Institute for Physics, South China Normal University（华南师范大学前沿物理研究所，广东-香港量子物质联合实验室，广东省量子工程与量子材料重点实验室）
The imaginary-time evolution method is widely known to be efficient for obtaining the ground state in quantum many-body problems on a classical computer. A recently proposed quantum imaginary-time evolution method (QITE) faces problems of deep circuit depth and difficulty in the implementation on noisy intermediate-scale quantum (NISQ) devices. In this study, a nonlocal approximation is developed to tackle this difficulty. We found that by removing the locality condition or local approximation (LA), which was imposed when the imaginary-time evolution operator is converted to a unitary operator…
Starting from a time-dependent Schrödinger equation, stationary states of 3D central potentials are obtained. An imaginary-time evolution technique coupled with the minimization of energy expectation value, subject to the orthogonality constraint leads to ground and excited states. The desired diffusion equation is solved by means of a finite-difference approach to produce accurate wave functions, energies, probability densities and other expectation values. Applications in case of 3D isotropic harmonic oscillator, Morse as well the spiked harmonic oscillator are made. Comparison with literatu…
We introduce a method to solve the MaxCut problem efficiently based on quantum imaginary time evolution (QITE). We employ a linear Ansatz for unitary updates and an initial state involving no entanglement, as well as an imaginary-time-dependent Hamiltonian interpolating between a given graph and a subgraph with two edges excised. We apply the method to thousands of randomly selected graphs with up to fifty vertices. We show that our algorithm exhibits a 93% and above performance converging to the maximum solution of the MaxCut problem for all considered graphs. Our results compare favorably wi…
Quantum Imaginary Time Evolution on an Infinite 1D Chain
National Taiwan University（台湾大学）；Center for Theoretical Physics, National Taiwan University（台湾大学理论物理中心）；Department of Physics, National Taiwan University（台湾大学物理系）
Quantum simulation on emerging quantum hardware is a topic of intense interest. While many studies focus on computing ground state properties or simulating unitary dynamics of closed systems, open quantum systems are an interesting target of study owing to their ubiquity and rich physical behavior. However, their non-unitary dynamics are also not natural to simulate on digital quantum devices. Here, we report algorithms for the digital quantum simulation of the dynamics of open quantum systems governed by a Lindblad equation using adaptations of the quantum imaginary time evolution (QITE) algo…
Finite Imaginary-Time Evolution for Polynomial Unconstrained Binary Optimization
Sungkyunkwan University（成均馆大学）；Department of Semiconductor Convergence Engineering, Sungkyunkwan University（成均馆大学半导体融合工程学院）；IBM Quantum（IBM量子）；Institute for Convergence Research and Education in Advanced Technology, Yonsei University（延世大学先进融合研究与教育学院）；Department of Quantum Information, Yonsei University（延世大学量子信息系）；Department of Applied Statistics, Yonsei University（延世大学应用统计系）；Department of Statistics and Data Science, Yonsei University（延世大学统计与数据科学系）；Department of Chemistry, Yonsei University（延世大学化学系）
Quasiprobabilistic imaginary-time evolution on quantum computers
Institute for Quantum Computing, University of Waterloo（滑铁卢大学量子计算研究所）；Department of Physics and Astronomy, University of Waterloo（滑铁卢大学物理与天文学系）；Perimeter Institute for Theoretical Physics（理论物理前沿研究所）；Inria Paris（法国国家信息与自动化研究所巴黎分部）
A probabilistic imaginary-time evolution quantum algorithm for advection-diffusion equation: Explicit gate-level implementation and comparisons to quantum linear system algorithms
In this paper we reformulate the problem of pricing options in a quantum setting. Our proposed algorithm involves preparing an initial state, representing the option price, and then evolving it using existing imaginary time simulation algorithms. This way of pricing options boils down to mapping an initial option price to a quantum state and then simulating the time dependence in Wick's imaginary time space. We numerically verify our algorithm for European options using a particular imaginary time evolution algorithm as proof of concept and show how it can be extended to path dependent options…
Developing scalable quantum algorithms to study finite-temperature physics of quantum many-body systems has attracted considerable interest due to recent advancements in quantum hardware. However, such algorithms in their present form require resources that exceed the capabilities of current quantum computers except for a limited range of system sizes and observables. Here, we report calculations of finite-temperature properties including energies, static and dynamical correlation functions, and excitation spectra of spin Hamiltonians with up to four sites on five-qubit IBM Quantum devices. Th…
The accurate computation of Hamiltonian ground, excited, and thermal states on quantum computers stands to impact many problems in the physical and computer sciences, from quantum simulation to machine learning. Given the challenges posed in constructing large-scale quantum computers, these tasks should be carried out in a resource-efficient way. In this regard, existing techniques based on phase estimation or variational algorithms display potential disadvantages; phase estimation requires deep circuits with ancillae, that are hard to execute reliably without error correction, while variation…
Computing the ground-state properties of quantum many-body systems is a promising application of near-term quantum hardware with a potential impact in many fields. The conventional algorithm quantum phase estimation uses deep circuits and requires fault-tolerant technologies. Many quantum simulation algorithms developed recently work in an inexact and variational manner to exploit shallow circuits. In this work, we combine quantum Monte Carlo with quantum computing and propose an algorithm for simulating the imaginary-time evolution and solving the ground-state problem. By sampling the real-ti…
Directly in the thermodynamic limit, we show how to combine imaginary and real time evolution of tensor networks to efficiently and accurately find the nonequilibrium steady states (NESS) of one-dimensional dissipative quantum lattices governed by the Lindblad master equation. The imaginary time evolution first bypasses any highly correlated portions of the real-time evolution trajectory by directly converging to the weakly correlated subspace of the NESS, after which real time evolution completes the convergence to the NESS with high accuracy. We demonstrate the power of the method with the d…
A fast implementation of the quantum imaginary time evolution (QITE) algorithm called Fast QITE is proposed. The algorithmic cost of QITE typically scales exponentially with the number of particles it nontrivially acts on in each Trotter step. In contrast, a Fast QITE implementation reduces this to only a linear scaling. It is shown that this speed up leads to a quantum advantage when sampling diagonal elements of a matrix exponential, which cannot be achieved using the standard implementation of the QITE algorithm. Finally the cost of implementing Fast QITE for finite temperature simulations …
Grover's algorithm is an approximation of imaginary-time evolution
Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院物理研究所）；Centre for Quantum Science and Engineering, École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院量子科学与工程中心）；Quantum Computing Center, Keio University（庆应义塾大学量子计算中心）；School of Physical and Mathematical Sciences, Nanyang Technological University（南洋理工大学物理与数学科学学院）；Centre for Quantum Technologies, Nanyang Technological University（南洋理工大学量子技术中心）
