11 research outputs found
Error-mitigated fermionic classical shadows on noisy quantum devices
Efficiently estimating fermionic Hamiltonian expectation values is vital for simulating various physical systems. Classical shadow (CS) algorithms offer a solution by reducing the number of quantum state copies needed, but noise in quantum devices poses challenges. We propose an error-mitigated CS algorithm assuming gate-independent, time-stationary, and Markovian (GTM) noise. For n-qubit systems, our algorithm, which employs the easily prepared initial state vertical bar 0(n)> < 0(n)vertical bar assumed to be noiseless, efficiently estimates k-RDMs with (O) over tilde (kn(k)) state copies and (O) over tilde(root n) calibration measurements for GTM noise with constant fidelities. We show that our algorithm is robust against noise types like depolarizing, damping, and X-rotation noise with constant strengths, showing scalings akin to prior CS algorithms for fermions but with better noise resilience. Numerical simulations confirm our algorithm's efficacy in noisy settings, suggesting its viability for near-term quantum devices
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 -qubit CNOT circuit could be
parallelized to depth, with 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 , any -qubit
CNOT circuit can be parallelized to depth, with 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 -depth quantum synthesis, and one by Patel et al. for :
for the former, we reduce the need of ancillae by a factor of by
showing that additional qubits suffice to build -depth, size --- which is asymptotically optimal --- CNOT
circuits; for the later, we reduce the depth by a factor of to the
asymptotically optimal bound . 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 -qubit CNOT circuit can be
paralleled to depth with ancillas for -dimensional grid
structure. For the high dimensional grid topological structure in which every
quibit connects to other qubits, we achieves the asymptotically
optimal depth with only ancillas. We also consider the
synthesis without ancillas. We propose an algorithm uses at most 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