Simulation of quantum circuits
Methods, systems and apparatus for simulating quantum circuits including multiple quantum logic gates. In one aspect, a method includes the actions of representing the multiple quantum logic gates as functions of one or more classical Boolean variables that define a undirected graphical model with each classical Boolean variable representing a vertex in the model and each function of respective classical Boolean variables representing a clique between vertices corresponding to the respective classical Boolean variables; representing the probability of obtaining a particular output bit string from the quantum circuit as a first sum of products of the functions; and calculating the probability of obtaining the particular output bit string from the quantum circuit by directly evaluating the sum of products of the functions. The calculated partition function is used to (i) calibrate, (ii) validate, or (iii) benchmark quantum computing hardware implementing a quantum circuit.
1. A computer-implemented method for simulating a quantum circuit the method comprising:
receiving data representing the quantum circuit;
generating a simulation of the quantum circuit using the received data, comprising:
representing the quantum circuit as a product of unitary matrices corresponding to a sequence of clock cycles, wherein the unitary matrices represent quantum logic gates included in the quantum circuit;
representing a probability that a predefined bit string is output after application of a final clock cycle of the quantum circuit on a system of qubits as a partition function of a classical Ising model of interacting spins, wherein the partition function of the classical Ising model comprises a sum, over each configuration of spins in the classical Ising model, of an exponential function of an energy of the configuration of spins multiplied by an inverse of an imaginary temperature, the probability being dependent on the product of unitary matrices; and
calculating, without applying approximations, the partition function to determine the probability that the predefined bit string is output after application of the final clock cycle of the quantum circuit on the system of qubits, the calculating comprising:
applying an algorithm for performing exact inference on a graphical models that represents the classical Ising model, wherein nodes in the graphical model represent the spins in the classical Ising model and edges in the graphical model represent interactions between spins; and
determining one or more properties of the quantum circuit using the generated simulation.
2. The method of claim 1 , wherein exactly calculating the partition function without applying approximations comprises:
determining a treewidth of the graphical model that represents the classical Ising model; and
applying, using the determined treewidth, the algorithm for performing exact inference on the graphical models to calculate the partition function of the classical Ising model.
3. The method of claim 1 , wherein the algorithm for performing exact inference on the graphical models that represents the classical Ising model comprises a bucket elimination algorithm or a variable elimination algorithm.
4. The method of claim 1 , wherein the computational cost of the algorithm for performing exact inference on the graphical model that represents the classical Ising model is exponential in a treewidth of the graphical model that represents the classical Ising model.
5. The method of claim 1 , wherein the probability that a predefined bit string is output after application of a final clock cycle of the quantum circuit is equal to a path integral comprising multiple phase factors associated with paths in the path integral, wherein values of the phase factors are accumulated as a sum of discrete phase changes associated with the quantum logic gates in the quantum circuit.
6. The method of claim 1 , wherein the quantum circuit is configured to operate on qubits arranged as a two-dimensional lattice.
7. The method of claim 1 , wherein the quantum circuit comprises (i) controlled-phase, (ii) X 1/2 , (iii) Y 1/2 , and (iv) non-Clifford T quantum logic gates.
8. The method of claim 1 , wherein the quantum circuit comprises a depth of d clock cycles.
9. The method of claim 1 , wherein the imaginary temperature comprises
i
π
4
.
10. The method of claim 1 , wherein the probability that a predefined bit string is output after application of a final clock cycle of the quantum circuit on a system of qubits is given by
〈
x
❘
ψ
d
〉
=
∑
{
σ
t
}
∏
t
=
0
d
〈
σ
t
❘
"\[LeftBracketingBar]"
U
(
t
)
❘
"\[RightBracketingBar]"
σ
t
-
1
〉
,
❘
"\[LeftBracketingBar]"
σ
d
〉
=
❘
"\[LeftBracketingBar]"
x
〉
where |x represents the predefined bit string, |ψ d represents the state of the system of n qubits after application of the final clock cycle d of the quantum circuit, U (t) represents the product of unitary matrices corresponding to the sequence of clock cycles 1 to d, and |σ t =⊗ j=1 n |σ j t with the assignments of σ j t =±1 corresponding to the states 0 and 1 of the j-th qubit, respectively.
11. The method of claim 5 , wherein the partition function of the classical Ising model at imaginary temperature is given by
〈
x
❘
ψ
d
〉
=
2
-
G
/
2
∑
s
exp
(
i
π
4
H
S
(
x
)
)
where's represents configurations of spins included in the classical Ising model, represents the
i
π
4
represents the inverse of the imaginary temperature, and
i
π
4
H
S
(
x
)
represents the energy of the configuration of spins s multiplied by the inverse of the imaginary temperature, wherein
i
π
4
H
S
(
x
)
also represents phase factors associated with each path in the path integral and 2 −G/2 represents absolute values of probability amplitudes associated with the paths in the path integral.
12. The method of claim 1 , wherein determining one or more properties of the quantum circuit using the generated simulation comprises using the calculated partition function to determine the performance of the quantum circuit.
13. The method of any one of claim 1 , wherein determining one or more properties of the quantum circuit using the generated simulation comprises using the calculated partition function to (i) calibrate, (ii) validate, or (iii) benchmark quantum computing hardware implementing the quantum circuit.
14. A system comprising one or more computers and one or more storage devices storing instructions that are operable, when executed by the one or more computers, to cause the one or more computers to perform the operations of:
receiving data representing a quantum circuit;
generating a simulation of the quantum circuit using the received data, comprising:
representing the quantum circuit as a product of unitary matrices corresponding to a sequence of clock cycles, wherein the unitary matrices represent quantum logic gates included in the quantum circuit;
representing a probability that a predefined bit string is output after application of a final clock cycle of the quantum circuit on a system of qubits as a partition function of a classical Ising model of interacting spins, wherein the partition function of the classical Ising model comprises a sum, over each configuration of spins in the classical Ising model, of an exponential function of an energy of the configuration of spins multiplied by an inverse of an imaginary temperature, the probability being dependent on the product of unitary matrices; and
calculating, without applying approximations, the partition function to determine the probability that the predefined bit string is output after application of the final clock cycle of the quantum circuit on the system of qubits, the calculating comprising:
applying an algorithm for performing exact inference on a graphical models that represents the classical Ising model, wherein nodes in the graphical model represent the spins in the classical Ising model and edges in the graphical model represent interactions between spins; and
determining one or more properties of the quantum circuit using the generated simulation.
15. A non-transitory computer storage medium encoded with instructions that, when executed by one or more computers, cause the one or more computers to perform operations comprising:
receiving data representing a quantum circuit;
generating a simulation of the quantum circuit using the received data, comprising:
representing the quantum circuit as a product of unitary matrices corresponding to a sequence of clock cycles, wherein the unitary matrices represent quantum logic gates included in the quantum circuit;
representing a probability that a predefined bit string is output after application of a final clock cycle of the quantum circuit on a system of qubits as a partition function of a classical Ising model of interacting spins, wherein the partition function of the classical Ising model comprises a sum, over each configuration of spins in the classical Ising model, of an exponential function of an energy of the configuration of spins multiplied by an inverse of an imaginary temperature, the probability being dependent on the product of unitary matrices; and
calculating, without applying approximations, the partition function to determine the probability that the predefined bit string is output after application of the final clock cycle of the quantum circuit on the system of qubits, the calculating comprising:
applying an algorithm for performing exact inference on a graphical models that represents the classical Ising model, wherein nodes in the graphical model represent the spins in the classical Ising model and edges in the graphical model represent interactions between spins; and
determining one or more properties of the quantum circuit using the generated simulation.