Systems and methods for compressed sensing measurement of long-range correlated noise

1. A computer-implemented method for detecting a two-qubit correlated dephasing error, the method comprising:

accessing a signal of a quantum device, the quantum device including a plurality of qubits, wherein every qubit has a nonzero rate of dephasing, wherein some qubits have a nonzero rate of correlated dephasing, wherein the signal includes information about a matrix, the matrix including diagonal elements and off-diagonal elements, and wherein the off-diagonal elements of the matrix are 2s-sparse;

performing randomized measurements of the off-diagonal elements of the matrix by preparing entangled Greenberger-Horne-Zeilinger states (GHZ states) on random subsets of the plurality of qubits;

recovering the matrix based on a direct measurement of the diagonal elements of the matrix; and

estimating a two-qubit correlated dephasing error based on the recovered matrix.

2. The computer-implemented method of claim 1 , wherein the randomized measurements are based on noise spectroscopy and quantum sensing.

3. The computer-implemented method of claim 1 , wherein the recovered matrix includes a restricted isometry property.

4. The computer-implemented method of claim 1 , further comprising estimating a vector of decay rates that are dependent on the matrix.

5. The computer-implemented method of claim 4 , wherein the recovered matrix is further based on the estimated decay rate.

6. The computer-implemented method of claim 4 , further comprising estimating a relaxation time of the quantum device based on the estimated vector of decay rates.

7. The computer-implemented method of claim 4 , further comprising estimating a decoherence time of the quantum device based on the estimated vector of decay rates.

8. The computer-implemented method of claim 1 , further comprising detecting a long-range correlated dephasing error based on the recovered matrix.

9. A computer-implemented method for detecting a two-qubit correlated dephasing error, the method comprising:

accessing a signal of a quantum device, the quantum device including a plurality of qubits, wherein every qubit of the plurality of qubits has a nonzero rate of dephasing and wherein some qubits have a nonzero rate of correlated dephasing;

dephasing entangled states of the plurality of qubits based on performing Ramsey spectroscopy using entangled states of random subsets of qubits of the plurality of qubits;

measuring a linear function of a correlation matrix, where the correlation matrix corresponds to correlated Markovian dephasing between pairs of qubits of the plurality of qubits;

generating a first vector and a second vector, wherein a plurality of elements of the first vector and of the second vector are randomly chosen; and

estimating a decay rate based on the first vector and the second vector.

10. The computer-implemented method of claim 9 , further comprising generating a restricted isometry property-based recovery matrix, wherein the recovery matrix is further based on the estimated decay rate.

11. The computer-implemented method of claim 10 , further comprising detecting a long-range correlated dephasing error based on the recovery matrix.

12. The computer-implemented method of claim 9 , further comprising estimating a relaxation time of the quantum device based on the estimated decay rate.

13. The computer-implemented method of claim 9 , further comprising estimating a decoherence time of the quantum device based on the estimated decay rate.

14. A system for detecting a two-qubit correlated dephasing error, the system comprising:

a processor; and

a memory, including instructions stored thereon, which when executed by the processor, cause the system to:

access a signal of a quantum device, the signal including a plurality of qubits, wherein every qubit has a nonzero dephasing rate;

generate a matrix C in R n×n , wherein its off-diagonal part is 2s sparse and wherein the matrix C is based on the accessed signal; and

detect a long-range correlated dephasing error based on the generated matrix C.

15. The system of claim 14 , wherein the instructions, when executed by the processor, further cause the system to:

perform randomized measurements based on:

estimating (b−a) T C(b−a) for any a≠b in {0,1}n;

choosing a sequence of random vectors r(1), . . . , r(m)in {1,0,−1}n; and

estimating ΦC=(r (1){circumflex over ( )}T Cr (1) ), . . . , (r (m){circumflex over ( )}T Cr (m) , where Φ:R n×n →R m .

16. The system of claim 15 , wherein the instructions, when executed by the processor, further cause the system to:

estimate a decay rate δ ab based on any a≠b in {0,1} n ; and

estimate a decoherence time of the quantum device based on the decay rate Γ ab .

17. The system of claim 16 , wherein the instructions, when executed by the processor, further cause the system to:

recover Matrix C based on:

measuring a plurality of diagonal elements of matrix C directly; and

determining an off-diagonal element of matrix C based on using 1 -regularized least-squares regression.