Bayesian quantum circuit fidelity estimation
Methods, systems and apparatus for estimating the fidelity of a quantum computing system. In one aspect, a method includes defining one or more random quantum circuits, wherein a noisy experimental implementation of each random quantum circuit is approximated by a depolarizing channel with respective polarization parameter; generating, for each defined random quantum circuit and by the quantum computing system, a set of experimental data, wherein data items in the set of experimental data comprise measured bit strings corresponding to experimental implementations of the random quantum circuit; determining, for each of the one or more random quantum circuits, an estimate of the respective polarization parameter, comprising maximizing a log-likelihood of the polarization parameter conditioned on the respective set of experimental data using series inversion; and determining an estimate of the fidelity of the quantum computing system based on the determined estimates of respective polarization parameters.
1 . A method for estimating the fidelity of a quantum computing system, the method comprising:
defining one or more random quantum circuits, wherein a noisy experimental implementation of each random quantum circuit is approximated by a depolarizing channel with respective polarization parameter;
generating, for each defined random quantum circuit and by the quantum computing system, a set of experimental data, wherein data items in the set of experimental data comprise measured bit strings corresponding to experimental implementations of the random quantum circuit;
determining, for each of the one or more random quantum circuits, an estimate of the respective polarization parameter, comprising:
computing, for each measured bit string in the respective set of experimental data, an output probability as a weighted combination of (i) a probability that an ideal implementation of the random quantum circuit produces a measurement eigenstate corresponding to the measured bit string and (ii) a uniform probability over all possible bit strings, wherein weights of the weighted combination are determined based on the polarization parameter;
forming a log-likelihood of the polarization parameter based on the computed output probabilities; and
maximizing the log-likelihood of the polarization parameter using series inversion; and
determining, using the determined estimates of respective polarization parameters, an estimate of the fidelity of the quantum computing system.
2 . The method of claim 1 , wherein determining an estimate of the respective polarization parameter comprises:
defining a new variable as equal to DP U (Z k )−1, where D represents Hilbert space dimension and P U (Z k ) represents a probability P U that an ideal implementation of the random quantum circuit U produces a measurement eigenstate corresponding to the k-th measured bit string Z k in the respective set of experimental data; and
substituting the new variable into a first equation for the first derivative of the log-likelihood of the polarization parameter conditioned on the respective set of experimental data to obtain an infinite series representation of the first equation.
3 . The method of claim 2 , wherein maximizing the log-likelihood of the polarization parameter conditioned on the respective set of experimental data using series inversion comprises:
computing a solution to the infinite series representation of the first equation for the first derivative of the log-likelihood of the polarization parameter conditioned on the respective set of experimental data using series inversion.
4 . The method of claim 1 , wherein generating a set of experimental data for a defined random quantum circuit comprises, repeatedly, for a predetermined number of times:
initializing a quantum computing system qubit register in an initial state;
applying the defined random quantum circuit to the initial state to generate an evolved state; and
measuring the evolved state to obtain a bit string.
5 . The method of claim 1 , further comprising determining a variance of the estimate of the respective polarization parameter by computing a second derivative of the log-likelihood of the polarization parameter conditioned on the respective set of experimental data.
6 . The method of claim 1 , wherein outputs of experimental implementations of the one or more random quantum circuits are approximated by a Porter-Thomas distribution.
7 . The method of claim 1 , wherein the one or more quantum circuits comprise random quantum circuits that operate on a same number of qubits and have a same circuit depth.
8 . The method of claim 1 , wherein determining, using the determined estimates of respective polarization parameters, an estimate of the fidelity of the quantum computing system comprises:
computing an average estimate of the polarization parameter; and
determining an estimate of the fidelity of the quantum computing system using the average estimate of the polarization parameter.
9 . The method of claim 8 , wherein the estimate of the fidelity F of the quantum computing system is given by
F
=
p
+
(
1
-
p
)
/
D
,
where p represents the average estimate of the polarization parameter, D=2 n represents Hilbert space dimension, and n represents a number of qubits on which the defined one or more random quantum circuits operate.
10 . The method of claim 8 , further comprising calculating an estimate of Pauli error rate of the quantum computing system using the average estimate of the polarization parameter.
11 . The method of claim 10 , wherein the estimate of Pauli error rate r Pauli of the quantum computing system is given by
r
Pauli
=
(
1
-
p
)
(
1
-
1
D
2
)
where p represents the average estimate of the polarization parameter, D=2 n represents Hilbert space dimension, and n represents a number of qubits on which the defined one or more random quantum circuits operate.
12 . The method of claim 1 , further comprising determining one or more properties of the quantum computing system using the determined estimate of the fidelity of quantum computing system.
13 . The method of claim 1 , further comprising:
determining one or more adjustments to quantum hardware control parameters based on the determined estimate of the fidelity; and
implementing the determined one or more adjustments to perform quantum computations using quantum computing hardware.
14 . An apparatus comprising:
one or more classical processors; and
quantum computing hardware in data communication with the one or more classical processors;
wherein the apparatus is configured to perform operations comprising:
defining one or more random quantum circuits, wherein a noisy experimental implementation of each random quantum circuit is approximated by a depolarizing channel with respective polarization parameter;
generating, for each defined random quantum circuit and by the quantum computing system, a set of experimental data, wherein data items in the set of experimental data comprise measured bit strings corresponding to experimental implementations of the random quantum circuit;
determining, for each of the one or more random quantum circuits, an estimate of the respective polarization parameter, comprising:
computing, for each measured bit string in the respective set of experimental data, an output probability as a weighted combination of (i) a probability that an ideal implementation of the random quantum circuit produces a measurement eigenstate corresponding to the measured bit string and (ii) a uniform probability over all possible bit strings, wherein weights of the weighted combination are determined based on the polarization parameter;
forming a log-likelihood of the polarization parameter based on the computed output probabilities; and
maximizing the log-likelihood of the polarization parameter using series inversion; and
determining, using the determined estimates of respective polarization parameters, an estimate of the fidelity of the quantum computing system.