METHOD OF MEASURING A RESPONSE FUNCTION OF A QUANTUM SYSTEM USING A QUANTUM COMPUTER
The method can include, for each one of one or more paths along which the quantum system may evolve from an initial state to a final state due to one or more actions of an effect: performing a plurality of iterations, each iteration including: preparing the initial state of the quantum system in a system register of the quantum computer; applying a sequence of operators interspersed with time evolutions in a permutation specific to the corresponding path, wherein the operators are one or more operators corresponding to the one or more actions of the effect and an operator corresponding to the action of an observable of the quantum system, the operators are applied by block encoding, and the time evolutions each correspond to one or more of the one or more time delays.
1 . A method of measuring a response function of a quantum system using a quantum computer, the method comprising:
performing a plurality of iterations for each one of one or more paths along which the quantum system may evolve from an initial state to a final state due to one or more actions of an effect, each iteration including:
preparing the initial state of the quantum system in a system register of the quantum computer, including applying a preparation circuit;
encoding possible values of one or more time delays as initial superpositions in corresponding ones of one or more auxiliary registers of the quantum computer;
applying a sequence of operators interspersed with time evolutions in a permutation specific to the corresponding path, wherein the operators are one or more operators corresponding to the one or more actions of the effect and an operator corresponding to the action of an observable of the quantum system, the time evolutions each correspond to one or more of the time delays, and the application of each time evolution is multiplexed on all of the one or more auxiliary registers associated to the corresponding one or more time delays;
applying a Hermitian conjugate of the preparation circuit to the system register;
measuring the system register;
applying a Fourier transform to the one or more auxiliary registers; and
measuring the one or more auxiliary registers; and
building one or more histograms based on the measurements acquired from each one of the plurality of iterations, and obtaining the response function from the one or more histograms.
2 . The method of claim 1 wherein the operators are applied by block encoding.
3 . The method of claim 1 wherein the plurality of iterations of the method are repeated for more than one path contributing to the response function, wherein, in the different repetitions, the permutation is different.
4 . The method of claim 1 wherein each iteration further includes determining whether the measurement of the system register corresponds to the initial state, wherein said building the histogram includes integrating the measurement of the one or more auxiliary registers to the histogram contingent upon said measurement of the system register corresponding to the initial state.
5 . The method of claim 1 wherein said building the histogram includes normalizing the histogram based on the measurements acquired in each one of the iterations.
6 . The method of claim 1 wherein the observable corresponds to an operator Â, the quantum system corresponds to a Hamiltonian Ĥ, there are a number D of action(s) of the effect, the effect corresponds to an operator {circumflex over (V)}, there are up to 2 D paths, the paths being paths in Liouville space, said encoding includes encoding D time delays t k between D actions of {circumflex over (V)} and one action of  in the permutation specific to the corresponding path, and the time evolutions are in the form exp(±iĤt).
7 . The method of claim 6 wherein said measuring the one or more auxiliary registers includes acquiring a binary encoding of D frequencies, and wherein said building the histogram includes building the histogram of the D frequencies based on the repeated acquisitions of the binary encoding, until the histogram converges to the response function within a level of statistical representativity.
8 . The method of claim 6 wherein when applying the operators and time evolutions, the different time evolutions are associated to different ones of the time delays t k and to different values of a variable k, and the operators are associated to said different values of the variable k in accordance with the permutation, and said applying the correlation function circuit includes executing the following routine:
i
)
set
k
=
1
;
ii) apply time evolution associated to a current value of k, controlled on all auxiliary registers which encode the corresponding time delay;
iii) apply block encoding of one of the operators associated to the current value of k:
iv
)
set
k
=
k
+
1
;
v) if k=D+1, apply the time evolution associated to current value of k, controlled on all auxiliary registers which encode the corresponding time delay, else go to ii).
9 . The method of claim 8 wherein, in different repetitions of the method corresponding to different ones of the paths, the operators are associated to different values of k.
10 . The method of claim 1 wherein the quantum system corresponds to a Hamiltonian represented in a Hilbert space, and the response function lies in a subspace of the Hilbert space, wherein said initial state of the quantum system and said operators are represented in the subspace of the Hilbert space.
11 . A method of measuring a response function of a quantum system using a quantum computer, the method comprising:
for each one of one or more paths along which the quantum system may evolve from an initial state to a final state due to one or more actions of an effect:
performing a plurality of iterations, each iteration including:
preparing the initial state of the quantum system in a system register of the quantum computer;
sampling, using a Monte Carlo process, one or more distributions of possible values of a number of time delays, the number of time delays corresponding to a number of the one or more actions, and thereby isolating a value of each one of the time delays;
performing a Hadamard test on the value of each one of the time delays, including:
applying a first Hadamard gate to an ancilla qubit, thereby preparing the ancilla qubit into a superposition;
applying a sequence of operators interspersed with time evolutions in a permutation specific to the corresponding path, wherein the operators are one or more operators corresponding to the one or more actions of the effect and an operator corresponding to the action of an observable of the quantum system, the time evolutions each correspond to one or more of the one or more time delays, and the application of each time evolution is controlled on the ancilla register;
if a number of the one or more actions of the effect is odd, applying a Hermitian conjugate of a phase gate on the ancilla qubit;
applying a second Hadamard gate on the ancilla qubit; and
measuring the ancilla qubit, thereby sampling a single random variable associated with the values of the delays from a function with a number of variables corresponding to the number of the one or more paths;
building a time series of the single random variables obtained from the plurality of iterations; and
performing a Fourier transform of the time series; and
obtaining the response function from a result of the Fourier transform performed for each one of the one or more paths.
12 . The method of claim 11 wherein said applying a sequence of operators includes applying the operators by block encoding.
13 . The method of claim 11 wherein the plurality of iterations of the method are repeated for more than one path contributing to the response function, wherein, in the different repetitions, the permutation is different.
14 . The method of claim 11 wherein the observable corresponds to an operator Â, the quantum system corresponds to a Hamiltonian Ĥ, there are a number D of action(s) of the effect, the effect corresponds to an operator {circumflex over (V)}, there are up to 2 D paths, the paths being paths in Liouville space, and the time evolutions are in the form exp(±iĤt).
15 . The method of claim 14 wherein when applying the operators and time evolutions, the different time evolutions are associated to different ones of the time delays t k and to different values of a variable k, and the operators are associated to said different values of the variable k in accordance with the permutation, and said applying the correlation function circuit includes executing the following routine:
i
)
set
k
=
1
;
ii) apply time evolution associated to a current value of k;
iii) apply block encoding of one of the operators associated to the current value of k;
iv
)
set
k
=
k
+
1
;
v) if k=D+1, apply the time evolution associated to current value of k, else go to ii).
16 . The method of claim 14 wherein, in different repetitions of the method corresponding to different ones of the paths, the operators are associated to different values of k.
17 . The method of claim 11 wherein the quantum system corresponds to a Hamiltonian represented in a Hilbert space, and the response function lies in a subspace of the Hilbert space, wherein said initial state of the quantum system and said operators are represented in the subspace of the Hilbert space.