Artificial thermal bath
Methods, systems, and apparatus for designing an artificial quantum thermal bath. In one aspect, an apparatus includes a primary quantum system; an auxiliary quantum system comprising a plurality of modes and that interacts with the primary quantum system, wherein: the auxiliary system interacts with the primary quantum system through one or more couplings, wherein the one or more couplings are controllable; a steady state of the auxiliary system, a response function of the auxiliary quantum system, an auxiliary system operator, and a spectrum of the auxiliary quantum system are determined to meet a respective condition; and the auxiliary system is controllable to relax the primary quantum system to an equilibrium state using a controllable function of the auxiliary system operator.
1. An apparatus, comprising:
a primary quantum system;
an auxiliary quantum system forming an artificial thermal quantum bath for the primary quantum system and comprising one or more resonators operating in a plurality of auxiliary quantum system modes, wherein the auxiliary quantum system interacts with the primary quantum system, wherein:
the auxiliary quantum system interacts with the primary quantum system through one or more couplings, wherein the one or more couplings are controllable;
a steady state of the auxiliary quantum system, a response function of the auxiliary quantum system, an auxiliary quantum system operator, and a frequency spectrum of the auxiliary quantum system are characterized as functions of tunable physical parameters of the auxiliary quantum system; and
the auxiliary quantum system is controllable to relax the primary quantum system to an equilibrium state using a controllable function of the auxiliary quantum system operator.
2. The apparatus of claim 1 , wherein the auxiliary quantum system interacts with an environment external to the primary quantum system and the auxiliary quantum system.
3. The apparatus of claim 1 , wherein dynamics of the auxiliary quantum system are controllable to relax the primary quantum system to the equilibrium state using the controllable function of the auxiliary quantum system operator.
4. The apparatus of claim 3 , wherein the dynamics of the auxiliary quantum system are controlled such that the auxiliary quantum system is an artificial thermal quantum bath for the primary quantum system.
5. The apparatus of claim 4 , wherein the equilibrium state of the primary quantum system is described by a Gibbs state e −βH s /Z where β −1 represents a tunable temperature of the primary quantum system, H S represents the primary quantum system Hamiltonian and Z represents a partition function which plays role of a normalizing constant.
6. The apparatus of claim 1 , wherein the controllable function of the auxiliary quantum system operator is a two-time correlation function of the auxiliary quantum system operator.
7. The apparatus of claim 1 , wherein the auxiliary quantum system comprises one or more multi-mode resonators, the plurality of auxiliary quantum system modes comprise resonator modes, and the spectrum of the auxiliary quantum system is a function of a set of the tunable physical parameters of the auxiliary quantum system, the tunable physical parameters comprising (i) a frequency and amplitude of a drive signal used to drive the one or more multi-mode resonators, and (ii) a leakage rate of the resonator modes.
8. The apparatus of claim 7 , wherein the set of tunable physical parameters are tuned using a controllable two-time correlation function.
9. The apparatus of claim 1 , wherein dynamics of the auxiliary quantum system are controlled such that
lim
t
→
∞
[
γ
α
β
(
-
ω
,
t
)
=
e
-
β
ω
γ
β
α
(
ω
,
t
)
]
is satisfied within a first accuracy, where t represents time, γ αβ (−ω, t) represents the response function of the auxiliary quantum system with α,β representing eigenvalues of a primary quantum system—thermal quantum bath tunable Hamiltonian, ω represents auxiliary quantum system frequency and β −1 represents a tunable temperature of the primary quantum system.
10. The apparatus of claim 1 , wherein dynamics of the auxiliary quantum system are controlled such that
γ
α
α′
(
ω
,
t
)
=
δ
α
α
′
e
β
2
ω
f
(
t
)
for
ω
∈
[
ω
min
,
ω
max
]
is satisfied within a second accuracy, where γ αα′ (ω, t)represents the response function of the auxiliary quantum system with α, α′ representing eigenvalues of a primary quantum system—thermal quantum bath interaction Hamiltonian, δ αα′ represents a Kronecker delta of the eigenvalues, β −1 represents a tunable temperature of the primary quantum system, ω represents auxiliary quantum system frequency, ƒ(t) represents a time-dependent real-valued function that provides a time dependent modulation of frequency characteristic of the response function of the auxiliary quantum system and ω min , ω max represent minimum and maximum frequency values of the auxiliary quantum system.
11. The apparatus of claim 10 , wherein dynamics of the auxiliary quantum system are controlled such that
γ
α
α′
(
ω
,
t
)
=
δ
α
α
′
e
β
2
ω
f
(
t
)
for
ω
∈
[
ω
min
,
ω
max
]
is satisfied within the second accuracy according to a set of values of response functions γ αα′ (ω, t), where γ αα′ (ω, t) represents the response function of the auxiliary quantum system with α, α′ representing eigenvalues of a primary quantum system—thermal quantum bath interaction Hamiltonian, δ αα′ represents a Kronecker delta of the eigenvalues, β −1 represents a tunable temperature of the primary quantum system, ω represents auxiliary quantum system frequency, ƒ(t) represents a time-dependent real-valued function that provides a time dependent modulation of frequency characteristic of the response function of the auxiliary quantum system and ω min , ω max represent minimum and maximum frequency values of the auxiliary quantum system.
12. The apparatus of claim 11 , wherein the minimum and maximum frequency values ω min and ω max are chosen such that energy scales of the primary quantum system correspond to auxiliary system frequencies that are included in the interval [ω min , ω max ].
13. The apparatus of claim 1 , wherein the auxiliary quantum system enables an analog realization of Quantum Metropolis Sampling.
14. The apparatus of claim 1 , wherein the auxiliary quantum system serves as a thermal drive for a hybrid quantum-classical annealer.
15. The apparatus of claim 1 , wherein the auxiliary quantum system comprises one or more multi-mode resonators.
16. The apparatus of claim 15 , wherein the primary quantum system comprises one or more qubits that are respectively coupled to a respective multi-mode resonator.
17. The apparatus of claim 15 , wherein the primary quantum system comprises one or more qubits that are collectively coupled to a multi-mode resonator.
18. A method, comprising:
identifying a primary quantum system;
identifying an auxiliary quantum system as a quantum bath, the auxiliary quantum system comprising one or more resonators operating in a plurality of auxiliary quantum system modes, wherein the auxiliary quantum system interacts with the primary quantum system;
determining that a steady state, response function and frequency spectrum of the auxiliary quantum system are each characterized as functions of tunable physical parameters of the auxiliary quantum system meet a respective condition;
approximating a thermalization condition for the primary quantum system using the response function and the spectrum of the auxiliary quantum system; and
controlling the auxiliary quantum system based on the approximated thermalization condition to achieve a range of tunable effective temperatures for the primary quantum system.
19. The method of claim 18 , wherein the auxiliary quantum system interacts with an environment external to the primary quantum system and the auxiliary quantum system.
20. The method of claim 18 , wherein controlling the auxiliary quantum system comprises controlling dynamics of the auxiliary quantum system.
21. The method of claim 20 , wherein controlling the dynamics of the auxiliary quantum system based on the approximated thermalization condition comprises tuning a set of physical parameters comprising (i) an auxiliary quantum system frequency, (ii) an auxiliary quantum system density of modes, (iii) a frequency and amplitude of a drive signal, and (iv) a leakage rate of resonator modes, and tuning a strength of a coupling between the primary quantum system and the auxiliary quantum system.
22. The method of claim 21 , wherein the set of physical parameters and the auxiliary quantum system modes are tuned in order to achieve a controllable two-time correlation function.
23. The method of claim 21 , wherein the auxiliary quantum system frequency is a member of a frequency range that corresponds to an effective energy scale of the primary quantum system.
24. The method of claim 21 , wherein the strength of the coupling between the primary quantum system and the auxiliary quantum system corresponds to a thermalization speed.
25. The method of claim 18 , further comprising using the auxiliary quantum system as an artificial thermal quantum bath for the primary quantum system.