Objective function estimation for solving optimization problems in gate-model quantum computers

Quantum computers provide a valuable resource to solve computational problems. The maximization of the objective function of a computational problem is a crucial problem in gate-model quantum computers. The objective function estimation is a high-cost procedure that requires several rounds of quantum computations and measurements. Here, we define a method for objective function estimation of arbitrary computational problems in gate-model quantum computers. The proposed solution significantly reduces the costs of the objective function estimation and provides an optimized estimate of the state of the quantum computer for solving optimization problems.


A.1 Estimation Error
The actual objective function values and the related parameter values serve illustration purposes.
In Fig. A.1(a), a distribution of a set S C i (z) 6 i=1 of observed objective function valuesC i (z) are depicted at a particular reference objective function valueC R (z). In Fig. A.1(b), the values of Φ(C R (z) ,C (z)) are depicted in function of the absolute value of the difference ofC R (z) andC i (z), i = 1, . . . , 6.

A.2 Notations
The notations of the manuscript are summarized in Table A.1. θ Refers to the collection θ 1 , . . . , θ N tot , where θ j is a continuous parameter.
|θ Quantum state of the quantum computer dominated by computational basis states with a high value of an objective function C.

QG
Quantum gate structure of the quantum computer.

D QG
Depth of the quantum circuit QG of the quantum computer.

Measurement.
N tot Total number of quantum gates of the QG quantum gate structure of the quantum computer.

U j
A jth unitary of the QG quantum gate structure of the quantum computer, j = 1, . . . , N tot .
|ψ 0 Initial state of the quantum computer.
|ψ 0 Initial state of the quantum computer.
f (θ ) Objective function of the quantum computer, f (θ ) = θ |C | θ , where C is a classical objective function of a computational problem.
where B j is a set of Pauli operators.

B j
Set of Pauli operators associated with a jth unitary U j of the QG quantum gate structure, j = 1, . . . , N tot .
N (ϕ j ) Total number of occurrences of gate parameter value ϕ j in the QG quantum circuit of the quantum computer.
Estimate of function f (θ ) at R * physical measurement rounds.
Estimate of function f (θ ) at R (κ) imaginary measurement rounds in the postprocessing.

Number of measurement rounds.
R * Total number of physical measurements rounds.
where R * is the total number of physical measurements, and R (κ) ≥ R * is the "imaginary" measurement rounds of the post-processing.

ξ κ
A quantity that measures the squared difference of the objective function values , where C r,(i) is the reference objective function in the ith physical measurement round, while C (i) is the objective function of the ith round.
C (z) Averaged objective function.
Objective function associated to the ith measurement round.

|M|
Total number of required measurements to get the estimate |θ * New quantum state produced by the quantum computer.

P (θ )
Prediction for the selection of the new value of θ for the QG quantum gate structure.
n Number of measurement gates in a round of M.
C 0 (z) Averaged objective function at R * rounds.
C 0 (z) Cumulative objective function at R * rounds.
A ϕ gate parameter associated to the (i, j)-th gate of the segmented QG circuit.
A E Objective function extension algorithm.
A D Quantum gate structure segmentation algorithm.
A L f Quantum gate parameter randomization algorithm, where L is the application level of A f .

Rule generation algorithm.
W Discrete wavelet transform.

W E (z)
Extended transformed objective function.
w (l) A parameter of W . Number of transformed objective function values at a given level l, l ≥ 1, expressed as w (l) = 4 + 3 (l − 1).

P E
Sub-procedure 1 of Algorithm 1 (A E ).

W * (z)
A maximized value of the results of W operators, Objective function extension factor. n t Number of classes selected for the segmentation of the ϕ gate parameters of the QG structure of the quantum computer in algorithm A D .
H k Entropy function associated to the kth class, k = 1, . . . , n t in algorithm A D .
f ( φ ) Objective function of the segmentation algorithm A D of the QG quantum gate structure.
φ l Gate segmentation parameter for to the classification of the ϕ gate parameters into lth and (l + 1)-th classes, 0 ≤ φ l ≤ χ, where χ is an upper bound on the gate parameters of the quantum computer.
Number of occurrences of ϕ i in the QG structure of the quantum computer.
Pr (N (ϕ i )) Probability distribution, Pr (N (ϕ i )) = N(ϕ i ) N tot , where N tot is the total number of quantum gates in the QG quantum circuit of the quantum computer, ∑ N tot i=1 Pr (N (ϕ i )) = 1. ω i Sum of probability distributions.
C QG ∈ ϕ j Classification of gate parameter ϕ j into C QG . ε φ * Error associated to the gate parameter segmentation algorithm A D .
Optimal rule with highest leverage coefficient.