A method for amplitude estimation in NISQ (Noisy Intermediate-Scale Quantum) computers

Abstract

An embodiment relates to a method for estimating the amplitude of a unitary operator U within an error ε by using a quantum processor configurable to implement the unitary operator U on a quantum circuit, which has a maximum depth S and can perform the unitary operator no more than D times in a single execution. A schedule of iterations n=1 to N based on the error ε and the number D is determined. Each iteration n is characterized by a schedule parameter kn, where kn≦D for all n and k n grows at a sub-exponential rate. Iterations n can be performed sequentially. In each iteration, the quantum processor calculates the unitary operator U by k n The quantum circuit is configured to execute the algorithm by sequentially applying it to the quantum circuit, and the non-quantum processor then estimates the amplitude of the unitary operator based on the resulting measured states.

Claims

1. 1. A method of estimating an amplitude of a unitary operator U to within an error ε by using a quantum processor configurable to implement the unitary operator U on a quantum circuit, the quantum circuit being capable of performing the unitary operator no more than a number D of times in a single execution, where D<O(1 / ε), the method comprising: determining a schedule of iterations n=1 to N based on the error ε and the number D, each iteration n having a schedule parameter k ｎ and k for all n ｎ ≦D, k ｎ grows at a sub-exponential rate and T<O(1 / ε ２ ) and [0010] and performing said iterations n sequentially, The unitary operator U is k ｎ configuring the quantum processor to sequentially apply a plurality of quantum circuits to the quantum circuit; said quantum processor executing said quantum circuit; measuring a resulting state of the quantum circuit; and and and a non-quantum processor estimating an amplitude of the unitary operator U based on the measured resulting states.

2. 2. The method of claim 1, wherein T<O(1 / ε ２ ) × 50 / 100.

3. 3. The method of claim 2, wherein T<O(1 / ε ２ ) × 20 / 100.

4. 2. The method of claim 1 , Ｎ =D, the method.

5. 10. The method of claim 1 , [0025] That is, the method.

6. 2. The method of claim 1 , ｎ is α where α is a constant and greater than zero.

7. The method of claim 6, wherein α∈(0,2].

8. 7. The method of claim 6, [0030] That is, the method.

9. 7. The method of claim 6, [0045] That is, the method.

10. 7. The method of claim 6, wherein α is T<O(1 / ε ２ )in [0050] It is predetermined that:

11. 7. The method of claim 6, wherein increasing α increases ε and decreases the execution time of the schedule, and decreasing α decreases ε and increases the execution time of the schedule.

12. The method of claim 1 , wherein one or more of the n iterations are performed multiple times.

13. 2. The method of claim 1, wherein for each iteration n, the quantum processor converts the unitary operator U into k ｎ times for the quantum circuit A ｎ , A being the depth of the unitary operator U.

14. 10. The method of claim 1 , wherein the maximum circuit depth of the quantum circuit is less than 50.

15. 10. The method of claim 1, wherein the quantum processor is comprised within a NISQ computer with less than 500 qubits.

16. 2. The method of claim 1, wherein D=5.

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