Quantum circuits for matrix trace estimation
Systems and methods for operating a quantum system are described. A controller of a quantum system can generate a command signal. The quantum system can include quantum hardware having a plurality of qubits. An interface of the quantum system can control the quantum hardware based on the command signal to generate a random state vector represented by the plurality of qubits. The random state vector can include a specific number of independent entries. The interface can control the quantum hardware to determine moments of a matrix based on the random state vector. The controller can be further configured to output the moments of the matrix to a computing device to estimate a trace of the matrix using the moments.
1 . An apparatus comprising:
a controller configured to generate a command signal;
quantum hardware including at least a first set of qubits, a second set of qubits, and a plurality of Hadamard gates; and
an interface connected to the controller and the quantum hardware, the interface being configured to convert the command signal received from the controller into a quantum operation to control the quantum hardware to:
generate a random state vector represented by a plurality of qubits, wherein the random state vector comprises a specific number of independent entries of independent states of the random state vector, and the random state vector is a superposition of multiple states of the random state vector having different Hamming weights;
determine moments of a matrix based on the random state vector, wherein elements of the matrix are inaccessible to the apparatus; and
the controller being further configured to output the moments of the matrix to a computing device to estimate a trace of the matrix using the moments.
2 . The apparatus of claim 1 , wherein the specific number of independent entries is four entries.
3 . The apparatus of claim 1 , wherein the matrix corresponds to a Laplacian of simplices of a specific order in a simplicial complex, and a determination of Betti numbers of the simplicial complex is based on the estimated trace.
4 . The apparatus of claim 1 , wherein the matrix is a Hermitian matrix.
5 . The apparatus of claim 1 , wherein the matrix is an n×n matrix representing a dataset of n data points.
6 . The apparatus of claim 1 , wherein the matrix is an n×n matrix, and the quantum circuit is configured to generate the random state vector by randomly sampling a column of a 2{circumflex over ( )}n×2{circumflex over ( )}n Hadamard matrix.
7 . The apparatus of claim 1 , wherein the quantum hardware comprises a quantum t-design circuit including a layer of parallel Hadamard gates followed by a set of Toffoli gates, the interface is configured to control the quantum t-design circuit to output pseudo-random states that are indistinguishable from states drawn from a random Haar measure to sample the random state vector.
8 . The apparatus of claim 1 , wherein the interface is configured to control the quantum hardware to:
determine a quantum state that represents an application of the matrix to the random state vector;
determine a complex conjugate of the random state vector; and
determine an inner product between the quantum state and the complex conjugate to determine the moments.
9 . The apparatus of claim 1 , wherein the matrix corresponds to a Laplacian of simplices of a specific order in a simplicial complex, and the quantum circuit is configured to:
determine a quantum state that represents an application of the Laplacian to the random state vector; and
determine a norm of the quantum state to determine the moments.
10 . The apparatus of claim 1 , wherein estimation of the trace comprises averaging the moments over a number of samples used for generating the random state vector.
11 . A system comprising:
a first computing device configured to process data encoded in binary data;
a second computing device configured to be in communication with the first computing device, the second computing device being configured to process data encoded in qubits, the second computing device comprises:
a controller configured to generate a command signal;
quantum hardware including at least a first set of qubits, a second set of qubits, and a plurality of Hadamard gates; and
an interface connected to the controller and the quantum hardware, the interface being configured to convert the command signal received from the controller into a quantum operation to control the quantum hardware to:
generate a random state vector represented by a plurality of qubits, wherein the random state vector comprises a specific number of independent entries of independent states of the random state vector, and the random state vector is a superposition of multiple states of the random state vector having different Hamming weights;
determine moments of a matrix based on the random state vector, wherein elements of the matrix are inaccessible to the apparatus; and
the controller being further configured to output the moments of the matrix to the first computing device to estimate a trace of the matrix using the moments.
12 . The system of claim 11 , wherein the specific number of independent entries is four entries.
13 . The system of claim 11 , wherein the matrix corresponds to a Laplacian of simplices of a specific order in a simplicial complex, and a determination of Betti numbers of the simplicial complex is based on the estimated trace.
14 . The system of claim 11 , wherein the matrix is a Hermitian matrix.
15 . The system of claim 11 , wherein the matrix is an n×n matrix representing a dataset of n data points.
16 . The system of claim 11 , wherein the matrix is an n×n matrix, and the quantum circuit is configured to generate the random state vector by randomly sampling a column of a 2{circumflex over ( )}n×2{circumflex over ( )}n Hadamard matrix.
17 . The system of claim 11 , wherein the second computing device comprises a quantum t-design circuit including a layer of parallel Hadamard gates followed by a set of Toffoli gates, and the interface is configured to control the quantum t-design circuit to output pseudo-random states that are indistinguishable from states drawn from a random Haar measure to sample the random state vector.
18 . The system of claim 11 , wherein the second computing device is configured to:
determine a quantum state that represents an application of the matrix to the random state vector;
determine a complex conjugate of the random state vector; and
determine an inner product between the quantum state and the complex conjugate to determine the moments.
19 . The system of claim 11 , wherein the matrix corresponds to a Laplacian of simplices of a specific order in a simplicial complex, and the second computing device is configured to:
determine a quantum state that represents an application of the Laplacian to the random state vector; and
determine a norm of the quantum state to determine the moments.
20 . The system of claim 11 , wherein the first computing device is configured to determine an average of the moments over a number of samples used for the generation of the random state vector to estimate the trace.
21 . A method of operating a quantum system, the method comprising:
receiving, by a controller of a quantum system, an instruction;
generating, by the controller of the quantum system, a command signal based on the instruction;
converting, by an interface of the quantum system, the command signal into a quantum operation; and
based on the quantum operation, controlling, by the interface of the quantum system, quantum hardware of the quantum system including a plurality of Hadamard gates to:
generate a random state vector represented by a plurality of qubits, the random state vector comprising a specific number of independent entries of independent states of the random state vector, and the random state vector is a superposition of multiple states of the random state vector having different Hamming weights;
determine moments of a matrix based on the random state vector, wherein elements of the matrix are inaccessible to the apparatus; and
outputting, by the controller of the quantum system, the moments of the matrix to a computing device to estimate a trace of the matrix using the moments.
22 . The method of claim 21 wherein the specific number of independent entries is four entries.
23 . The method of claim 21 , wherein the matrix corresponds to a Laplacian of simplices of a specific order in a simplicial complex, and a determination of Betti numbers of the simplicial complex is based on the estimated trace.
24 . The method of claim 21 , wherein an estimation of the trace comprises determining an average of the moments over a number of samples used for the generation of the random state vector.
25 . The method of claim 21 , wherein the matrix is a Hermitian matrix.