Quantum machine learning (QML) is a rapidly rising research field that incorporates ideas from quantum computing and machine learning to develop emerging tools for scientific research and improving data processing. How to efficiently control or manipulate the quantum system is a fundamental and vexing problem in quantum computing. It can be described as learning or approximating a unitary operator. Since the success of the hybrid-based quantum machine learning model proposed in recent years, we investigate to apply the techniques from QML to tackle this problem. Based on the Choi–Jamiołkowski isomorphism in quantum computing, we transfer the original problem of learning a unitary operator to a min–max optimization problem which can also be viewed as a quantum generative adversarial network. Besides, we select the spectral norm between the target and generated unitary operators as the regularization term in the loss function. Inspired by the hybrid quantum-classical framework widely used in quantum machine learning, we employ the variational quantum circuit and gradient descent based optimizers to solve the min-max optimization problem. In our numerical experiments, the results imply that our proposed method can successfully approximate the desired unitary operator and dramatically reduce the number of quantum gates of the traditional approach. The average fidelity between the states that are produced by applying target and generated unitary on random input states is around 0.997.

With the enormous variety of applications, machine learning has already impacted modern life [

Based on this fact, quite a lot of quantum machine learning models that harness the quantum advantages have been proposed in recent years. Quantum support vector machine (QSVM) and its physical implementation that runs on a small-scale quantum device result in an exponential speed-up over the corresponding classical one [

In this work, relying on the advantages of the hybrid-based quantum machine learning model, we intend to exploit it for a unitary learning problem that is an essential task in quantum computing. One reason why learning unitary matrices matters a lot is that the dynamics of a quantum system can be described by a unitary transformation, exploring the efficient way to implement the unitary transformation is fundamental. Since we are still in the early stages of quantum computing with the limitation of quantum resources for computation, it is necessary to investigate how to control or manipulate the quantum system efficiently. Besides, various tasks will benefit from the results of the study on learning unitary transformation. For instance, unitary matrix compiling, that is the task of ‘compiling’ a known unitary transformation into a quantum circuit constructed by a sequence of gates chosen from the specific quantum gate set. A large amount of existing works has been focused on how to implement unitary transformations efficiently. Such works provide theoretic analysis of approximate error and gate or time complexity for implementing target unitary transformation [

We organized this paper as follows. In Section 2, we formalize the learning problem of unitary transformation. In Section 3, we give a brief introduction of the techniques in hybrid quantum-classical framework and quantum generative adversarial networks (QGAN). In Section 4, we propose a method by which a quantum machine learning model can be used for solving the formulized unitary learning problem. In Section 5, we provide the numerical experiments which demonstrate the advantages of the proposed model. We also introduce the background of quantum computing in the appendix.

In quantum mechanics, we have two pictures for the dynamics of a quantum system. One is the Hamiltonian picture; another is the unitary operator picture. In the Hamiltonian picture, given the Hamiltonian, a Hermitian operator, that is the total energy function of the closed system, we can describe the evolution of the system by Schrödinger equation as follow,

It can be verified that the solution of

Because the Hamiltonian

In the following, our discussion is mainly under the unitary operator picture. As shown in

Plenty of methods were proposed in past years. Lloyd [

In this section, we will introduce the hybrid-based quantum machine learning model. The hybrid quantum-classical framework is derived from the quantum approximate optimization algorithm (QAOA) [

where we are able to program the

By calculating the derivative, in terms of parameters

The framework of hybrid-based model is shown in

The works listed above form the problem as a non-convex optimization problem, such as minimizing or maximizing a composite function

where the observable

where

By virtue of the success of QAOA, the non-convex optimization of the loss function

Variational quantum circuit (VQC), known as the parametrized quantum circuit, plays the core role in the hybrid quantum-classical framework. It can be viewed as the quantum black-box model that is capable of approximating any given unitary operator with a small error like neural networks in classical machine learning [

where

In subgraph (a) of

Different from the methods introduced to solve the problem of learning a unitary transformation in Section 2, we reformulate the problem from another perspective. According to Choi–Jamiołkowski isomorphism in quantum information theory [

where

Since the unitary transformation _{T}

where the quantum state

The learning problem of _{T}_{T}

There are many choices for

Given two quantum state

Trace distance:

It can be viewed as a generalization of the total variation distance on probability distribution that satisfies the symmetry and triangle inequality. It is also related to the maximum probability of distinguish different quantum state. The variational form of trace distance is shown in the second line of

Fidelity:

Although the fidelity is not a metric, it has many good properties, such as it is invariant unitary transformation, the value of fidelity lies within 0 and 1. It also can be interpreted as the angle between states on a unit sphere. If

Quantum Optimal Transport Distance:

It is the quantum extension of classical optimal transport distance [

Because the quantum optimal transport distance can naturally be implemented by hybrid quantum-classical framework on quantum device, we select quantum optimal transport distance as

Given a unitary operator

Trace norm (

The Frobenius norm (

The Spectral norm (

where

There are many nice properties of Schatten

The related work employed the square of Frobenius norm as the loss function for learning unitary transformation [

Since the min-max problem of

As the spectral norm of matrix

In this section, we provide the experiments of applying the quantum machine learning model for learning the unitary transformation. We apply the quantum machine learning model discussed above to the task of learning the unitary transformation of one dimensional Heisenberg spin model which is also considered in reference [_{T}

In which the Hamiltonian

where _{j}

In

We select fidelity between the generated and target quantum states to evaluate the performance of our model. In _{T}

Learning unitary transformation is an important and vexing task in quantum computing, which is related to controlling a quantum system or implementing a quantum algorithm with fewer resources. In this work, we investigate the use of promising techniques from quantum machine learning for learning a unitary transformation of a quantum system. Instead of the related works which formulate the learning problem as minimizing the norms between target and generated unitary operator, we express the problem as a quantum generative adversarial network with regularization term from the other perspective based on Choi–Jamiołkowski isomorphism. Comparing the trace norm and Frobenius norm used in related works, we add an intuitively stronger norm, spectral norm, as the regularization term to the loss function. Our numerical experiments demonstrate that the operator generated by our proposed model can successfully approximate the desired target unitary operator. The average spectral norm error of 10-replication runs is 0.1, and the average fidelity between the states produced by applying target and generated operators on the random input is around 0.997. Meanwhile, compared to the traditional method using product formulas for Hamiltonian simulation, our proposed model can significantly reduce the number of quantum gates for implementation. There are some potential applications of our proposed model such as providing help for assisting us in implementing quantum algorithms or compiling quantum circuits.

Thanks for constructive suggestion and helpful discussions from professor Xiaodi Wu.

Instead of storing a certain state in one classical bit, a qubit, the counterpart of classical bit, is in a superposition of two basic state _{i}

The widely used model for quantum computation is the quantum circuit model that is constructed by a sequence of reversible quantum logic gates. In quantum mechanics, the time evolution of a quantum system can be described by a unitary transformation. Thus, the quantum logic gates are equivalent to the unitary transformations in the quantum circuit model. The followings are commonly used one-qubit and multiple-qubits quantum gates,

Puali Gates,

Rotation Gates,

Entangling Gates,

Controlled-NOT gate,

Ising XX Coupling Gates,

An example of quantum circuit is shown in

The classical data stored in quantum states cannot be directly read out. To extract the data, we have to perform the quantum measurement on the state. Due to the measurement involve the distraction from the exterior environment, the system will collapse after the measurement which is an irreversible action. The operation of measurement can be described by a group of measurement operator