Quantum Phase Recognition via Quantum Kernel Methods

The application of quantum computation to accelerate machine learning algorithms is one of the most promising areas of research in quantum algorithms. In this paper, we explore the power of quantum learning algorithms in solving an important class of Quantum Phase Recognition (QPR) problems, which are crucially important in understanding many-particle quantum systems. We prove that, under widely believed complexity theory assumptions, there exists a wide range of QPR problems that cannot be efficiently solved by classical learning algorithms with classical resources. Whereas using a quantum computer, we prove the efficiency and robustness of quantum kernel methods in solving QPR problems through Linear order parameter Observables. We numerically benchmark our algorithm for a variety of problems, including recognizing symmetry-protected topological phases and symmetry-broken phases. Our results highlight the capability of quantum machine learning in predicting such quantum phase transitions in many-particle systems

Tensor-network-assisted variational quantum algorithm

Near-term quantum devices generally suffer from shallow circuit depth and hence limited expressivity due to noise and decoherence. To address this, we propose tensor-network-assisted parametrized quantum circuits, which concatenate a classical tensor-network operator with a quantum circuit to effectively increase the circuit's expressivity without requiring a physically deeper circuit. We present a framework for tensor-network-assisted variational quantum algorithms that can solve quantum many-body problems using shallower quantum circuits. We demonstrate the efficiency of this approach by considering two examples of unitary matrix-product operators and unitary tree tensor networks, showing that they can both be implemented efficiently. Through numerical simulations, we show that the expressivity of these circuits is greatly enhanced with the assistance of tensor networks. We apply our method to two-dimensional Ising models and one-dimensional time-crystal Hamiltonian models with up to 16 qubits and demonstrate that our approach consistently outperforms conventional methods using shallow quantum circuits.Comment: 12 pages, 8 figures, 37 reference

Complexity analysis of weakly noisy quantum states via quantum machine learning

Quantum computers capable of fault-tolerant operation are expected to provide provable advantages over classical computational models. However, the question of whether quantum advantages exist in the noisy intermediate-scale quantum era remains a fundamental and challenging problem. The root of this challenge lies in the difficulty of exploring and quantifying the power of noisy quantum states. In this work, we focus on the complexity of weakly noisy states, which we define as the size of the shortest quantum circuit required to prepare the noisy state. To analyze the complexity, we propose a quantum machine learning (QML) algorithm that exploits the intrinsic-connection property of structured quantum neural networks. The proposed QML algorithm enables efficiently predicting the complexity of weakly noisy states from measurement results, representing a paradigm shift in our ability to characterize the power of noisy quantum computation

Optimal Space-Depth Trade-Off of CNOT Circuits in Quantum Logic Synthesis

Due to the decoherence of the state-of-the-art physical implementations of quantum computers, it is essential to parallelize the quantum circuits to reduce their depth. Two decades ago, Moore et al. demonstrated that additional qubits (or ancillae) could be used to design "shallow" parallel circuits for quantum operators. They proved that any $n$-qubit CNOT circuit could be parallelized to $O(\log n)$ depth, with $O(n^2)$ ancillae. However, the near-term quantum technologies can only support limited amount of qubits, making space-depth trade-off a fundamental research subject for quantum-circuit synthesis. In this work, we establish an asymptotically optimal space-depth trade-off for the design of CNOT circuits. We prove that for any $m\geq0$, any $n$-qubit CNOT circuit can be parallelized to $O\left(\max \left\{\log n, \frac{n^{2}}{(n+m)\log (n+m)}\right\} \right)$ depth, with $O(m)$ ancillae. We show that this bound is tight by a counting argument, and further show that even with arbitrary two-qubit quantum gates to approximate CNOT circuits, the depth lower bound still meets our construction, illustrating the robustness of our result. Our work improves upon two previous results, one by Moore et al. for $O(\log n)$-depth quantum synthesis, and one by Patel et al. for $m = 0$: for the former, we reduce the need of ancillae by a factor of $\log^2 n$ by showing that $m=O(n^2/\log^2 n)$ additional qubits suffice to build $O(\log n)$-depth, $O(n^2/\log n)$ size --- which is asymptotically optimal --- CNOT circuits; for the later, we reduce the depth by a factor of $n$ to the asymptotically optimal bound $O(n/\log n)$. Our results can be directly extended to stabilizer circuits using an earlier result by Aaronson et al. In addition, we provide relevant hardness evidences for synthesis optimization of CNOT circuits in term of both size and depth.Comment: 25 pages, 5 figures. Fixed several minor typos and a mistake about CNOT+Rz circui

Optimization of CNOT circuits on topological superconducting processors

We focus on optimization of the depth/size of CNOT circuits under topological connectivity constraints. We prove that any $n$-qubit CNOT circuit can be paralleled to $O(n)$ depth with $n^2$ ancillas for $2$-dimensional grid structure. For the high dimensional grid topological structure in which every quibit connects to $2\log n$ other qubits, we achieves the asymptotically optimal depth $O(\log n)$ with only $n^2$ ancillas. We also consider the synthesis without ancillas. We propose an algorithm uses at most $2n^2$ CNOT gates for arbitrary connected graph, considerably better than previous works. Experiments also confirmed the performance of our algorithm. We also designed an algorithm for dense graph, which is asymptotically optimal for regular graph. All these results can be applied to stabilizer circuits

Theory of Quantum Generative Learning Models with Maximum Mean Discrepancy

The intrinsic probabilistic nature of quantum mechanics invokes endeavors of designing quantum generative learning models (QGLMs) with computational advantages over classical ones. To date, two prototypical QGLMs are quantum circuit Born machines (QCBMs) and quantum generative adversarial networks (QGANs), which approximate the target distribution in explicit and implicit ways, respectively. Despite the empirical achievements, the fundamental theory of these models remains largely obscure. To narrow this knowledge gap, here we explore the learnability of QCBMs and QGANs from the perspective of generalization when their loss is specified to be the maximum mean discrepancy. Particularly, we first analyze the generalization ability of QCBMs and identify their superiorities when the quantum devices can directly access the target distribution and the quantum kernels are employed. Next, we prove how the generalization error bound of QGANs depends on the employed Ansatz, the number of qudits, and input states. This bound can be further employed to seek potential quantum advantages in Hamiltonian learning tasks. Numerical results of QGLMs in approximating quantum states, Gaussian distribution, and ground states of parameterized Hamiltonians accord with the theoretical analysis. Our work opens the avenue for quantitatively understanding the power of quantum generative learning models.Comment: 28 pages, 9 figure

Quantum logic synthesis for Satisfiability Problems

To demonstrate the advantage of quantum computation, many attempts have been made to attack classically intractable problems, such as the satisfiability problem (SAT), with quantum computer. To use quantum algorithms to solve these NP-hard problems, a quantum oracle with quantum circuit implementation is usually required. In this manuscript, we first introduce a novel algorithm to synthesize quantum logic in the Conjunctive Normal Form (CNF) model. Compared with linear ancillary qubits in the implementation of Qiskit open-source framework, our algorithm can synthesize an m clauses n variables k-CNF with $O(k^2 m^2/n)$ quantum gates by only using three ancillary qubits. Both the size and depth of the circuit can be further compressed with the increase in the number of ancillary qubits. When the number of ancillary qubits is $\Omega(m^\epsilon)$ (for any $\epsilon > 0$), the size of the quantum circuit given by the algorithm is O(km), which is asymptotically optimal. Furthermore, we design another algorithm to optimize the depth of the quantum circuit with only a small increase in the size of the quantum circuit. Experiments show that our algorithms have great improvement in size and depth compared with the previous algorithms.Comment: 5 pages,3 figures, with Supplementar