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Non-linear quantum-classical scheme to simulate non-equilibrium strongly correlated fermionic many-body dynamics

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ABSTRACT

We propose a non-linear, hybrid quantum-classical scheme for simulating non-equilibrium dynamics of strongly correlated fermions described by the Hubbard model in a Bethe lattice in the thermodynamic limit. Our scheme implements non-equilibrium dynamical mean field theory (DMFT) and uses a digital quantum simulator to solve a quantum impurity problem whose parameters are iterated to self-consistency via a classically computed feedback loop where quantum gate errors can be partly accounted for. We analyse the performance of the scheme in an example case.

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Hybrid non-equilibrium DMFT simulation results when dynamically increasing the Hubbard tunneling matrix element v(t) from 0 to v0 as described in the main text.We choose U = 2v0, Trotter steps Δt = 0.04/v0 and couple the impurity site to N = 2 bath sites. (a) Impurity double occupation  as a function of time t: numerically exact solution (blue solid curve), solution with Trotter errors (+), solutions including gate errors of σMS = 0.1% (green dashed curve), σMS = 1% (yellow solid curve), and σMS = 10% (red solid curve). (b) Absolute value of the difference  between the imaginary parts of the lesser Green’s function without gate errors and with gate errors of σMS = 1%. Results of calculations with gate errors are obtained by averaging over 128 realizations of the setup.
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f3: Hybrid non-equilibrium DMFT simulation results when dynamically increasing the Hubbard tunneling matrix element v(t) from 0 to v0 as described in the main text.We choose U = 2v0, Trotter steps Δt = 0.04/v0 and couple the impurity site to N = 2 bath sites. (a) Impurity double occupation as a function of time t: numerically exact solution (blue solid curve), solution with Trotter errors (+), solutions including gate errors of σMS = 0.1% (green dashed curve), σMS = 1% (yellow solid curve), and σMS = 10% (red solid curve). (b) Absolute value of the difference between the imaginary parts of the lesser Green’s function without gate errors and with gate errors of σMS = 1%. Results of calculations with gate errors are obtained by averaging over 128 realizations of the setup.

Mentions: We analyze the performance of our simulation scheme by considering a simple example system24. We study the infinite-dimensional time-dependent Hubbard model (1) with constant onsite interaction U and tunneling matrix element v(t). The simulation starts in the half-filled paramagnetic atomic limit with tunneling v(t = 0) = 0, which is then dynamically ramped up to its final value v0 after quench time 1/4v0 and is kept at v0 until the final simulation time tmax is reached24 (setting ħ = 1). Such a sudden quench is representative of experimental ultracold atom dynamics3233 and also ultrafast dynamics probed in condensed matter systems19. The initial state of the system has a singly occupied impurity site in the completely mixed state of spin ↑ and spin ↓, and one half of the bath sites are doubly occupied and the other half empty (for explicit details, see ref. 24). In practice, we prepare the system in two pure fermion occupational number states, where one has the impurity in state /↑〉 and the other in state /↓〉, along with the bath states24. The results are then averaged over these two pure states. These initial number states are mapped onto product states of qubits via the Jordan-Wigner transformation (see Methods). The initial qubit configuration is that shown in Fig. 2, where . We emulate the operation of the quantum coprocessor by classically evaluating the quantum networks, and the classical exponential scaling limits our simulations to small systems. The self-consistency condition for the Bethe lattice calculated in the classical feedback loop is Λσ(t, t′) = v(t)Gσ(t, t′)v(t′), from which we obtain the SIAM coupling to bath p efficiently via a Cholesky decomposition , where * denotes complex conjugation (see Supplementary Material for details). The impurity site double occupancy obtained from the self-consistent hybrid simulation is compared to the exact result in Fig. 3a and shows that Trotter errors do not noticeably affect our results.


Non-linear quantum-classical scheme to simulate non-equilibrium strongly correlated fermionic many-body dynamics
Hybrid non-equilibrium DMFT simulation results when dynamically increasing the Hubbard tunneling matrix element v(t) from 0 to v0 as described in the main text.We choose U = 2v0, Trotter steps Δt = 0.04/v0 and couple the impurity site to N = 2 bath sites. (a) Impurity double occupation  as a function of time t: numerically exact solution (blue solid curve), solution with Trotter errors (+), solutions including gate errors of σMS = 0.1% (green dashed curve), σMS = 1% (yellow solid curve), and σMS = 10% (red solid curve). (b) Absolute value of the difference  between the imaginary parts of the lesser Green’s function without gate errors and with gate errors of σMS = 1%. Results of calculations with gate errors are obtained by averaging over 128 realizations of the setup.
© Copyright Policy - open-access
Related In: Results  -  Collection

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getmorefigures.php?uid=PMC5016988&req=5

f3: Hybrid non-equilibrium DMFT simulation results when dynamically increasing the Hubbard tunneling matrix element v(t) from 0 to v0 as described in the main text.We choose U = 2v0, Trotter steps Δt = 0.04/v0 and couple the impurity site to N = 2 bath sites. (a) Impurity double occupation as a function of time t: numerically exact solution (blue solid curve), solution with Trotter errors (+), solutions including gate errors of σMS = 0.1% (green dashed curve), σMS = 1% (yellow solid curve), and σMS = 10% (red solid curve). (b) Absolute value of the difference between the imaginary parts of the lesser Green’s function without gate errors and with gate errors of σMS = 1%. Results of calculations with gate errors are obtained by averaging over 128 realizations of the setup.
Mentions: We analyze the performance of our simulation scheme by considering a simple example system24. We study the infinite-dimensional time-dependent Hubbard model (1) with constant onsite interaction U and tunneling matrix element v(t). The simulation starts in the half-filled paramagnetic atomic limit with tunneling v(t = 0) = 0, which is then dynamically ramped up to its final value v0 after quench time 1/4v0 and is kept at v0 until the final simulation time tmax is reached24 (setting ħ = 1). Such a sudden quench is representative of experimental ultracold atom dynamics3233 and also ultrafast dynamics probed in condensed matter systems19. The initial state of the system has a singly occupied impurity site in the completely mixed state of spin ↑ and spin ↓, and one half of the bath sites are doubly occupied and the other half empty (for explicit details, see ref. 24). In practice, we prepare the system in two pure fermion occupational number states, where one has the impurity in state /↑〉 and the other in state /↓〉, along with the bath states24. The results are then averaged over these two pure states. These initial number states are mapped onto product states of qubits via the Jordan-Wigner transformation (see Methods). The initial qubit configuration is that shown in Fig. 2, where . We emulate the operation of the quantum coprocessor by classically evaluating the quantum networks, and the classical exponential scaling limits our simulations to small systems. The self-consistency condition for the Bethe lattice calculated in the classical feedback loop is Λσ(t, t′) = v(t)Gσ(t, t′)v(t′), from which we obtain the SIAM coupling to bath p efficiently via a Cholesky decomposition , where * denotes complex conjugation (see Supplementary Material for details). The impurity site double occupancy obtained from the self-consistent hybrid simulation is compared to the exact result in Fig. 3a and shows that Trotter errors do not noticeably affect our results.

View Article: PubMed Central - PubMed

ABSTRACT

We propose a non-linear, hybrid quantum-classical scheme for simulating non-equilibrium dynamics of strongly correlated fermions described by the Hubbard model in a Bethe lattice in the thermodynamic limit. Our scheme implements non-equilibrium dynamical mean field theory (DMFT) and uses a digital quantum simulator to solve a quantum impurity problem whose parameters are iterated to self-consistency via a classically computed feedback loop where quantum gate errors can be partly accounted for. We analyse the performance of the scheme in an example case.

No MeSH data available.