Ground-State Projection with Auxiliary Fields: Imaginary-Time and Simulation-Time Dynamics

Infinite projected entangled pair states (iPEPS), the tensor network ansatz for two-dimensional systems in the thermodynamic limit, already provide excellent results on ground-state quantities using either imaginary-time evolution or variational optimisation. Here, we show (i) the feasibility of real-time evolution in iPEPS to simulate the dynamics of an infinite system after a global quench and (ii) the application of disorder-averaging to obtain translationally invariant systems in the presence of disorder. To illustrate the approach, we study the short-time dynamics of the square lattice Heisenberg model in the presence of a bi-valued disorder field.

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THE DISSIPATIVE DOUBLE-WELL POTENTIAL: A MULTILEVEL SPIN HOPPING ANALYSIS

Modern Physics Letters B ◽

10.1142/s0217984991000411 ◽

1991 ◽

Vol 05 (05) ◽

pp. 351-356

Author(s):

H. DEKKER

Keyword(s):

Energy Spectrum ◽

Ground State ◽

Vibrational Relaxation ◽

Coupling Model ◽

Quantum Mechanical ◽

Time Dynamics ◽

Novel Treatment ◽

Model Hamiltonian ◽

Double Well Potential ◽

Ground State Doublet

A novel treatment is presented of the real-time dynamics of a quantum mechanical particle in a dissipative double-well potential at finite temperatures. The analysis is based on the bilinear coupling model Hamiltonian à la Zwanzig. The energy spectrum consists of a ladder of vibrational doublets. The usual truncation to the ground state doublet — à la Leggett et al. — is not required. The intra-doublet spin-boson dynamics is evaluated in the “noninteracting-blip approximation”. The inter-doublet vibrational relaxation gives rise to a stochastic hopping process.

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Adaptive Variational Quantum Imaginary Time Evolution Approach for Ground State Preparation

Advanced Quantum Technologies ◽

10.1002/qute.202100114 ◽

2021 ◽

pp. 2100114

Author(s):

Niladri Gomes ◽

Anirban Mukherjee ◽

Feng Zhang ◽

Thomas Iadecola ◽

Cai‐Zhuang Wang ◽

...

Keyword(s):

Time Evolution ◽

Ground State ◽

Imaginary Time ◽

State Preparation ◽

Evolution Approach

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Prime factorization using quantum variational imaginary time evolution

Scientific Reports ◽

10.1038/s41598-021-00339-x ◽

2021 ◽

Vol 11 (1) ◽

Author(s):

Raja Selvarajan ◽

Vivek Dixit ◽

Xingshan Cui ◽

Travis S. Humble ◽

Sabre Kais

Keyword(s):

Time Evolution ◽

Ground State ◽

Single Layer ◽

Imaginary Time ◽

Quantum Devices ◽

Prime Factorization ◽

Promising Alternative ◽

The Road ◽

Variational Techniques ◽

Shor's Algorithm

AbstractThe road to computing on quantum devices has been accelerated by the promises that come from using Shor’s algorithm to reduce the complexity of prime factorization. However, this promise hast not yet been realized due to noisy qubits and lack of robust error correction schemes. Here we explore a promising, alternative method for prime factorization that uses well-established techniques from variational imaginary time evolution. We create a Hamiltonian whose ground state encodes the solution to the problem and use variational techniques to evolve a state iteratively towards these prime factors. We show that the number of circuits evaluated in each iteration scales as $$O(n^{5}d)$$ O ( n 5 d ) , where n is the bit-length of the number to be factorized and d is the depth of the circuit. We use a single layer of entangling gates to factorize 36 numbers represented using 7, 8, and 9-qubit Hamiltonians. We also verify the method’s performance by implementing it on the IBMQ Lima hardware to factorize 55, 65, 77 and 91 which are greater than the largest number (21) to have been factorized on IBMQ hardware.

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Theory of variational quantum simulation

Quantum ◽

10.22331/q-2019-10-07-191 ◽

2019 ◽

Vol 3 ◽

pp. 191 ◽

Cited By ~ 15

Author(s):

Xiao Yuan ◽

Suguru Endo ◽

Qi Zhao ◽

Ying Li ◽

Simon C. Benjamin

Keyword(s):

Variational Principle ◽

Time Evolution ◽

Variational Principles ◽

Gibbs State ◽

Ritz Method ◽

Quantum Simulation ◽

Imaginary Time ◽

Unitary Evolution ◽

Time Dynamics ◽

Phase Alignment

The variational method is a versatile tool for classical simulation of a variety of quantum systems. Great efforts have recently been devoted to its extension to quantum computing for efficiently solving static many-body problems and simulating real and imaginary time dynamics. In this work, we first review the conventional variational principles, including the Rayleigh-Ritz method for solving static problems, and the Dirac and Frenkel variational principle, the McLachlan's variational principle, and the time-dependent variational principle, for simulating real time dynamics. We focus on the simulation of dynamics and discuss the connections of the three variational principles. Previous works mainly focus on the unitary evolution of pure states. In this work, we introduce variational quantum simulation of mixed states under general stochastic evolution. We show how the results can be reduced to the pure state case with a correction term that takes accounts of global phase alignment. For variational simulation of imaginary time evolution, we also extend it to the mixed state scenario and discuss variational Gibbs state preparation. We further elaborate on the design of ansatz that is compatible with post-selection measurement and the implementation of the generalised variational algorithms with quantum circuits. Our work completes the theory of variational quantum simulation of general real and imaginary time evolution and it is applicable to near-term quantum hardware.

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Variational quantum Boltzmann machines

Quantum Machine Intelligence ◽

10.1007/s42484-020-00033-7 ◽

2021 ◽

Vol 3 (1) ◽

Author(s):

Christa Zoufal ◽

Aurélien Lucchi ◽

Stefan Woerner

Keyword(s):

Numerical Simulation ◽

Ground State ◽

Gibbs State ◽

Discriminative Learning ◽

Imaginary Time ◽

Boltzmann Machines ◽

Learning Tasks ◽

State Approximation ◽

Near Term ◽

Actual Loss

AbstractThis work presents a novel realization approach to quantum Boltzmann machines (QBMs). The preparation of the required Gibbs states, as well as the evaluation of the loss function’s analytic gradient, is based on variational quantum imaginary time evolution, a technique that is typically used for ground-state computation. In contrast to existing methods, this implementation facilitates near-term compatible QBM training with gradients of the actual loss function for arbitrary parameterized Hamiltonians which do not necessarily have to be fully visible but may also include hidden units. The variational Gibbs state approximation is demonstrated with numerical simulations and experiments run on real quantum hardware provided by IBM Quantum. Furthermore, we illustrate the application of this variational QBM approach to generative and discriminative learning tasks using numerical simulation.

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THE HUBBARD-STRATONOVICH TRANSFORMATION AND THE HUBBARD MODEL

International Journal of Modern Physics B ◽

10.1142/s0217979291000493 ◽

1991 ◽

Vol 05 (06n07) ◽

pp. 937-976 ◽

Cited By ~ 11

Author(s):

SANDRO SORELLA

Keyword(s):

Wave Function ◽

Hubbard Model ◽

Ground State ◽

Trial Wave Function ◽

Many Body ◽

Imaginary Time ◽

Many Body Theory ◽

Straightforward Application ◽

Ground State Properties ◽

Body Theory

The Hubbard-Stratonovich transformation allows one to formulate the problem of calculating the ground state properties of a many-body theory as one of sampling a distribution. This distribution is constructed by propagating a trial wave function under the influence of a one-body time-dependent external field. However, a straightforward application of the Hubbard-Stratonovich transformation gives distributions which are not always positive definite for a generic trial wave function. In this work it is rigorously shown that, for the Hubbard model, in many cases the non-positiveness of this distribution is not important for reaching the infinite imaginary time limit, i.e., the ground state properties.

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