Variational quantum Gibbs state preparation

1. A method for preparing a Gibbs state in a qubit register of a quantum computer configured to implement one or more quantum gates, the method comprising:

applying the one or more quantum gates to one or more qubits of the qubit register to prepare a trial quantum state spanning the one or more qubits, the trial quantum state being defined as a function of parameters {right arrow over (θ)} and being selected to provide an initial estimate of the Gibbs state;

evaluating the Gibbs free energy of the trial quantum state, including expanding an entropy term Tr[σ({right arrow over (θ)})log σ({right arrow over (θ)})] as a Fourier series, according to Tr[ρlogρ]≈Σ m c m (1) Tr[ρcos(jρ)]+Σ m c m (2) Tr[ρsin(jρ)], where Tr is a trace operator, ρ is a distribution, ρ is a density matrix, c m (1) and c m (2) are Fourier coefficients, and j is an integer;

adjusting the parameter {right arrow over (θ)};

re-applying the one or more quantum gates to the one or more qubits to refine the trial quantum state according to the parameters {right arrow over (θ)} as adjusted; and

re-evaluating the Gibbs free energy of the trial quantum state as refined,

wherein said adjusting, re-applying, and re-evaluating are repeated iteratively so as to lower the Gibbs free energy of the trial quantum state, thereby forming the Gibbs state in the qubit register.

2. The method of claim 1 further comprising: computing a change in the Gibbs free energy pursuant to re-evaluating the Gibbs free energy of the trial quantum state as refined; and returning that trial quantum state as the Gibbs state if the change is below a predetermined threshold.

3. The method of claim 1 further comprising computing a gradient of the Gibbs free energy with respect to the parameters {right arrow over (θ)}, based on the Gibbs free energy as re-evaluated, wherein adjusting the parameters includes adjusting based on the gradient.

4. The method of claim 1 wherein adjusting the parameters includes adjusting without estimation of a gradient of the Gibbs free energy with respect to the parameters {right arrow over (θ)}.

5. The method of claim 1 wherein the quantum computer is one or more of a fault-tolerant quantum computer and a near-term quantum-computer device.

6. The method of claim 1 wherein the Gibbs free energy is defined as a difference Tr[σ({right arrow over (θ)})H]−kTTr[σ({right arrow over (θ)})log σ({right arrow over (θ)})] of energy and entropy terms, wherein operator H is a fixed Hamiltonian operator, distribution σ is a function of the parameters {right arrow over (θ)}, T is temperature, k is the Boltzmann constant, Tr represents a trace, and log represents a natural logarithm.

7. The method of claim 1 wherein expanding the entropy term Tr[σ({right arrow over (θ)})log σ({right arrow over (θ)})] as a Fourier series includes approximating the logarithm by a truncated Taylor series and converting the truncated Taylor series into a Fourier series.

8. The method of claim 1 wherein expanding the entropy term Tr[σ({right arrow over (θ)})log σ({right arrow over (θ)})] as a Fourier series includes computing the Fourier coefficients in a non-quantum computer coupled operatively to the quantum computer.

9. The method of claim 1 wherein expanding the entropy term Tr[σ({right arrow over (θ)})log σ({right arrow over (θ)})] as a Fourier series includes applying the one or more quantum gates to the one or more qubits to evaluate Fourier components of the Fourier series.

10. The method of claim 1 wherein expanding the entropy term Tr[σ({right arrow over (θ)})log σ({right arrow over (θ)})] as a Fourier series includes applying the one or more quantum gates to the one or more qubits to exponentiate a density matrix.

11. The method of claim 1 wherein evaluating and re-evaluating the Gibbs free energy includes estimating an energy term.

12. The method of claim 11 wherein estimating the energy term includes applying a Hadamard test and expanding the Hamiltonian operator as a linear combination of unitary operations.

13. The method of claim 1 wherein the trial quantum state as selected is a purified density matrix.

14. The method of claim 1 wherein the trial quantum state as selected is a Trotterized adiabatic state.

15. A quantum computer comprising:

a register including a plurality of qubits;

a modulator configured to implement one or more quantum-logic operations on the plurality of qubits;

a demodulator configured to output data based on a quantum state of the plurality of qubits;

a controller operatively coupled to the modulator and to the demodulator; and

computer memory holding instructions that cause the controller to:

apply the one or more quantum gates to one or more qubits of the qubit register to prepare a trial quantum state spanning the one or more qubits, the trial quantum state being defined as a function of parameters {right arrow over (θ)} and being selected to provide an initial estimate of a Gibbs state,

evaluate the Gibbs free energy of the trial quantum state, including expanding an entropy term Tr[σ({right arrow over (θ)})log σ({right arrow over (θ)})] as a Fourier series, according to Tr[ρlog ρ≈Σ m c m (1) Tr[ρcos(jρ)]+Σ m c m (2) Tr[ρsin(jρ)], where Tr is a trace operator, ρ is a distribution, ρ is a density matrix, c m (1) and c m (2) are Fourier coefficients, and j is an integer;

adjust the parameters {right arrow over (θ)};

re-apply the one or more quantum gates to the one or more qubits to refine the trial quantum state according to the parameters {right arrow over (θ)} as adjusted; and

re-evaluate the Gibbs free energy of the trial quantum state as refined,

wherein said adjusting, re-applying, and re-evaluating are repeated iteratively so as to lower the Gibbs free energy of the trial quantum state, thereby forming the Gibbs state in the qubit register.

16. The quantum computer of claim 15 wherein the instructions cause the controller to iteratively repeat adjustment of the parameters, re-application of the one or mor quantum gates to the one or more qubits to refine the trial quantum state, and re-evaluation of the Gibbs free energy of the trial quantum state so as to lower the Gibbs free energy of the trial quantum state, the method further comprising:

computing a change in the Gibbs free energy pursuant to re-evaluating the Gibbs free energy of the trial quantum state; and returning the trial quantum state as the Gibbs state if the change is below a predetermined threshold.

17. A method for preparing a Gibbs state in a qubit register of a quantum computer configured to implement one or more quantum gates, the method comprising:

applying the one or more quantum gates to one or more qubits of the qubit register to prepare a trial quantum state spanning the one or more qubits, the trial quantum state being defined as a function of parameters {right arrow over (θ)} and being selected to provide an initial estimate of the Gibbs state;

evaluating the Gibbs free energy of the trial quantum state;

adjusting the parameters {right arrow over (θ)};

re-applying the one or more quantum gates to the one or more qubits to refine the trial quantum state according to the parameters {right arrow over (θ)} as adjusted; and

re-evaluating the Gibbs free energy of the trial quantum state, as refined,

wherein said adjusting, re-applying, and re-evaluating are repeated iteratively so as to lower the Gibbs free energy of the trial quantum state, thereby forming the Gibbs state in the qubit register.

wherein the Gibbs free energy is defined as a difference of energy and entropy terms, and wherein evaluating and re-evaluating the Gibbs free energy includes:

expanding the entropy term Tr[σ({right arrow over (θ)})logσ({right arrow over (θ)})] as a Fourier series, according to Tr[ρlog ρ]≈Σ m c m (1) Tr[ρcos(jρ)]+Σ m c m (2) Tr[ρsin(jρ)], where Tr is a trace operator, ρ is a distribution, ρ is a density matrix, c m (1) and c m (2) are Fourier coefficients, and j is an integer, and

applying the one or more quantum gates to the one or more qubits to evaluate Fourier components of the Fourier series.

18. The method of claim 17 wherein expanding the entropy term as a Fourier series includes applying the one or more quantum gates to the one or more qubits to exponentiate a density matrix.