State Stabilization for Gate-Model Quantum Computers

09/03/2019
by   Laszlo Gyongyosi, et al.
0

Gate-model quantum computers can allow quantum computations in near-term implementations. The stabilization of an optimal quantum state of a quantum computer is a challenge, since it requires stable quantum evolutions via a precise calibration of the unitaries. Here, we propose a method for the stabilization of an optimal quantum state of a quantum computer through an arbitrary number of running sequences. The optimal state of the quantum computer is set to maximize an objective function of an arbitrary problem fed into the quantum computer. We also propose a procedure to classify the stabilized quantum states of the quantum computer into stability classes. The results are convenient for gate-model quantum computations and near-term quantum computers.

READ FULL TEXT VIEW PDF
POST COMMENT

Comments

There are no comments yet.

Authors

page 16

09/03/2019

Training Optimization for Gate-Model Quantum Neural Networks

Gate-based quantum computations represent an essential to realize near-t...
03/06/2018

Quantum Circuit Designs for Gate-Model Quantum Computer Architectures

The power of quantum computers makes it possible to solve difficult prob...
07/22/2021

Multiple Query Optimization using a Hybrid Approach of Classical and Quantum Computing

Quantum computing promises to solve difficult optimization problems in c...
03/11/2020

Optimizing High-Efficiency Quantum Memory with Quantum Machine Learning for Near-Term Quantum Devices

Quantum memories are a fundamental of any global-scale quantum Internet,...
01/21/2022

Short-Range Microwave Networks to Scale Superconducting Quantum Computation

A core challenge for superconducting quantum computers is to scale up th...
04/30/2022

Quantum Approximate Optimization Algorithm with Sparsified Phase Operator

The Quantum Approximate Optimization Algorithm (QAOA) is a promising can...
02/25/2022

Short Paper: Device- and Locality-Specific Fingerprinting of Shared NISQ Quantum Computers

Fingerprinting of quantum computer devices is a new threat that poses a ...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

1 Introduction

Quantum computers can make possible quantum computations for efficient problem solving [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. Gate-based quantum computations represent a way to construct gate-model quantum computers. In a gate-model quantum computer architecture, computations are implemented via sequences of unitary operations [12, 13, 14, 15, 22, 23, 24, 25, 26, 27, 28, 29, 30]. Gate-model quantum computers allow establishing experimental quantum computations in near-term architectures [1, 2, 3, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47]. Practical demonstrations of gate-model quantum computers have been already proposed [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15] and several physical-layer developments are currently in progress.

Finding a stable quantum state of a quantum computer is a challenge, since it requires precise unitaries that yield stable quantum evolutions in the quantum computer. The problem is further increased if the stable system state must be available for a pre-determined time or for a pre-determined number of running sequences. Particularly, the quantum state of a quantum computer subject to stabilization also coincides with the optimal quantum state. The optimal quantum state of a quantum computer maximizes a particular objective function of an arbitrary computational problem fed into the quantum computer. The problem therefore is to fix the quantum state of the quantum computer in the optimal state for an arbitrary number of running sequences that is determined by the actual environment or by the current problem. Another challenge connected to the problem of stabilization of the system state of a quantum computer is the classification of the sequences of the stabilized quantum states into stability-classes. Practically, a solution to these problems can be covered by an unsupervised learning method.

Here, we propose a method for the stabilization of an optimal quantum state of a quantum computer through an arbitrary number of running sequences. We define a solution that utilizes unsupervised learning algorithms to determine the stable quantum states of the quantum computer and to classify the stable quantum states into stability classes. The proposed results are useful for experimental gate-based quantum computations and near-term quantum computer architectures.

The novel contributions of our manuscript are as follows:

  1. We propose a method for the stabilization of an optimal quantum state of a quantum computer through an arbitrary number of running sequences.

  2. We define a solution that utilizes unsupervised learning algorithms to determine the stable quantum states of the quantum computer.

  3. We evaluate a solution to classify the stable system states into stability classes.

This paper is organized as follows. Section 2 provides the problem statement. Section 3 discusses the stabilization procedure of an optimal quantum state of a quantum computer. Section 4 defines an unsupervised learning method to find the stable quantum states and the stability classes of the stabilized quantum states. In Section 5, a numerical evaluation is proposed. Finally, Section 6 concludes with the results. Supplemental information is included in the Appendix.

2 Problem Statement

Let be the quantum gate structure of a gate-model quantum computer with a sequence of unitaries [12, 13, 14, 15] with an -length input system ,

(1)

where is the dimension (

=2 for a qubit system),

, and let

(2)

be the optimal system state of the quantum computer that maximizes a particular objective function ,

(3)

of an arbitrary problem fed into the quantum computer, where is the classical value of the objective function, while

is the gate parameter vector,

(4)

that identifies the unitaries, , of the quantum circuit of the quantum computer in the optimal state , such that an -th unitary, is as [13]

(5)

where is the gate parameter (real continuous variable) of unitary ,

is a generalized Pauli operator formulated by the tensor product of Pauli operators

[13, 14].

The aim is to stabilize the optimal state of the quantum computer through running sequences via unsupervised learning of the evolution of the unitaries in the quantum computer.

The running sequences refers to input systems fed into the input of the quantum computer, such that in an -th running sequence, , an -th input system, (defined as in (1)), is evolved via the sequence of the uniaries of the quatum computer. The running sequences identify an input system, , formulated via , -length quantum systems, as

(6)

where it is considered that the input systems are unentangled.

Let be the gate parameter vector associated with the stable system state ,

(7)

as

(8)

where is the gate parameter of unitary in the stabilized system state , such that the objective function value is stabilized into

(9)

For the sequences of the quantum computer, we define matrices and as

(10)

where identifies the quantum state of an -th running sequence of the quantum computer, while

(11)

where , identifies the stabilized quantum state of an -th sequence of the quantum computer.

The problem therefore is to find from that stabilizes the optimal state of the quantum computer through sequences as

(12)

where is a stabilizer matrix,

(13)

and

is the identity matrix.

The problems to be solved are therefore summarized as follows.

Problem 1

Find to construct (11) from (10) to stabilize the quantum computer in via for all running sequences.

Problem 2

Describe the stability of via unsupervised learning of the stability levels of the quantum states of .

The resolutions of Problems 1 and 2 are proposed in Theorems 1 and 2. The solution framework is defined via a stabilization procedure with an embedded stabilization algorithm (see Theorem 1), and via an classification algorithm that characterizes the stability class of the results of (see Theorem 2). Fig. 1 depicts the system model.

Figure 1: The framework for the stabilization of the optimal state of the quantum computer and the stability-class determination. In the running sequences, input systems are fed into the input of the quantum computer, in an -th running sequence, , an -th input system, , is evolved via the sequence of the uniaries of the quatum computer. The running sequences identify an input system (considering that the input systems are unentangled). The running sequences of the structure of the quantum computer produces , where . The stabilization procedure outputs , where , via an embedded stabilization algorithm that determines the stabilizer matrix. The stability-level of the resulting is determined via a classification algorithm . The and methods are realized as unsupervised learning.

3 Stabilization of the Optimal State of the Quantum Computer

Theorem 1

The matrix for the stabilization of the optimal state of the quantum computer via can be determined via the minimization of an objective function .

Proof. For an -th sequence of the quantum computer, define and as

(14)

and

(15)

respectively. These vectors formulate and as

(16)

and

(17)

respectively. Then, using equations (16) and (17) for the sequences, let be a sum defined as

(18)

where is the squared -norm, is the trace operator, is as given in equation (16), and is as in equation (17).

For the -th and -th sequences, , with and , let be defined as

(19)

where is a weight coefficient defined as

(20)

where and are nonzero parameters.

A sum is defined for the sequences of the quantum computer as

(21)

At a particular in equations (18) and (21), the stabilization of the optimal state of the quantum computer through the sequences can be reformulated via an objective function , subject to a minimization as

(22)

where is a regularization constant [31, 32]. The objective function therefore stabilizes the optimal state via the minimization of , while the term achieves stabilization between the sequences.

Then, let be the weight matrix formulated via the coefficients (20) with , and let be a diagonal matrix of the weight coefficients (20) with

(23)

such that

(24)

Using and , the objective function in equation (22) can be rewritten as

(25)

where is as

(26)

and

(27)

At a particular (16) and (26), the stabilizer matrix in equation (25) is evaluated via

(28)

where

is a diagonal matrix of eigenvalues

[31, 32].

Algorithm A.1 () gives the method for stabilizing the optimal state of the quantum computer.

Step 1. Set the number of sequences for the quantum state stabilization. Formulate (10) via gate parameter vectors , . Step 2. Set , and determine the weight coefficients via equation (20) for all and . Step 3. Set , , and (26). Step 4. Compute the stabilizer matrix via equation (28). Step 5. Output via equation (12) for the stabilization of the optimal quantum state via the stable state (7) through sequences of the quantum computer.
Algorithm 1 Stabilization of the Optimal State of the Quantum Computer

 

4 Learning the Stable Quantum State and Stability Class

Lemma 1

The stabilized sequences of the quantum computer can be determined via unsupervised learning.

Proof. Algorithm 1 with the objective function (25) can be used to formulate an unsupervised learning framework to find the stabilized unitaries. The steps are detailed in Procedure 1 ().

Step 1. Construct a training set of random gate parameters of the -structure of the quantum computer, as
(29)
where is a -dimensional random vector, , formulated as
(30)
where is the gate parameter of in , and is a random number. Step 2. Determine the stabilizer matrix via Algorithm 1. Step 3. Compute as
(31)
and set as
(32)
where is the mean of all training samples [32]. Step 4. For a given of an -th sequence, learn output as
(33)
Step 5. For an -th gate parameter , learn the -th output as
(34)
from which a statistical average for a given , , is
(35)
with difference as
(36)
Step 6. Repeat step 5 for all . Step 7. Repeat steps 1-5 for the sequences.
Procedure 2 Unsupervised Learning of Stable Quantum Evolutions

 

4.1 Learning the Sequence Stability of Stabilized Quantum States

Proposition 1

The stability of a given sequence can be characterized via stability levels. The sequence can be classified into stability classes from set ,

(37)

where , , is the -th stability class.

Theorem 2

The , stability class of a stabilized sequence, , of the quantum computer can be learned via quantities in the high-dimensional Hilbert space , where and

is a probability.

Proof. Since the gate parameters are stabilized, the gate parameters and of the -th and -th sequences must be correlated in the stable system state (7) of the quantum computer.

Let from equation (11) be the stabilized sequences, where is the stabilized gate parameter vector of an -th sequence of the quantum computer, and let be the set of all sequences of gate parameters as

(38)

For an -th stabilization class , a probabilistic classifier function [31, 37] can be defined as

(39)

The goal is to learn a function that maps any sequence to the correct stability class. Applying equation (39) on a given sequence , i.e., therefore maps to a given stability class via the classification of each gate parameter of the sequence.

Thus, an -th stabilized gate parameter of an -th sequence can be also classified into a particular stabilization class from (37). The , , classifier (39) is trained to classify [37] each of the gate parameters of , via outputting a corresponding probability that belongs to a given class. For a particular , the sum of the probabilities yields

(40)

for all .

Then, let be a weight parameter associated with a particular and -th class , defined as

(41)

which normalizes into the range of , .

For an -th sequence , a collection can be defined as

(42)

where .

From equations (39) and (42), the evolution of a particular sequence with respect to a -th class is defined as

(43)

Since the term (43) is a non-linear map, the problem of correlation analysis [31, 37] between the inner products of non-linear functions and can be reformulated via a kernel machine [34, 35, 36] as , which yields a distance in a high-dimensional Hilbert space . This distance in can therefore be used as a metric to describe the correlation between and .

Let be the input space and let be an arbitrary kernel machine, defined for a given via the kernel function

(44)

where

(45)

is a nonlinear map from to the high-dimensional reproducing kernel Hilbert space (RKHS) associated with . Without a loss of generality, , and we assume that the map in equation (45) has no inverse.

Then, for a and , let be the correlation identifier, as

(46)

Assuming that is a Gaussian kernel [34, 35, 36] in equation (46), for an -th gate parameter the kernel function is

(47)

where , while yields the -distance in ,

(48)

For a given and , an average is yielded as

(49)

where refers to the space of symmetric positive semi-definite matrices [35, 36, 37], while the inner products of and are represented in via as

(50)

The sequence is classified into a given class from set , as given in Algorithm 2 ().

Step 1. Let be the -th sequence of the quantum computer, with the stabilized gate parameters . Step 2. Define set of the stability classes via (37). Step 3. Select that identifies -th stability class , and learn function (46) using the kernel machine (44) for all , . Step 4. Determine . Step 5. Repeat steps 3-4 for all . Step 6. Determine . Step 7. Classify into stability class via the set as
where indexes the maximal in , while indexes the maximal in . Step 8. Repeat steps 1-8 for all . Step 9. Output the stability classes of the stabilized quantum states , of the quantum computer.
Algorithm 3 Learning the Classification of the Stabilized Quantum States of the Quantum Computer

 

5 Numerical Evaluation

5.1 System Stability

Let be the stabilized system state of the quantum computer formulated by output systems, , , as

(51)

with gate parameters , as given in (11).

Then, let be a target stabilized system of the quantum computer, as

(52)

with target gate parameters , as

(53)

where .

Then, let refer to the gate parameter of an -th unitary of an -th running sequence of the quantum computer, , , in the state , and let identify the target gate parameter in state .

Then, let be the relative entropy between and , as

(54)

where , and let be a function that returns the value of the relative entropy function for an -th running sequence as

(55)

Let be a target value for function (55), and let be the difference [33] between and (55), as

(56)

where is the derivative of .

Using (56), we define a stability parameter to quantify the variation of the stabilized system state of the -th running sequence of the quantum computer, as

(57)

For analytical purposes, let us assume that oscillates between a minimal value , and a maximal value , defined as

(58)

and

(59)

therefore can be rewritten as

(60)

where is a constant, set as

(61)

while is an expected value of (54), set as

(62)

while is the number of oscillations.

Therefore, (56) can be evaluated as

(63)

where , with .

Then, by using (63), the quantity in (57) is as

(64)

that identifies the inverse of the number of oscillations.

Therefore, (64) identifies the stability of the system state of the quantum computer in the -th running sequence if has the form of (60). For an arbitrary , the stability parameter is evaluated via (57). The high value of indicates that the stabilized system in (51) changes slowly. Particularly, if , where is a target value for , then the system state of the quantum computer is considered as stable.

The values of (60) and (64) for running sequences are depicted in Fig. 2.

Figure 2: The relative entropy values between the gate parameters of the stabilized system and target system for running sequences, , , . (a) . (b) . (c) . (d). Stability parameter for the different relative entropy values.

5.2 Gate Parameter Correlations

Let be the stabilized state of the quantum computer in the -th running sequence, with , and let be the target stabilized system state in the -th running sequence, with .

Then, let be a correlation coefficient [33] that measures the correlation of the gate parameters and of (51) and (11), defined as