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. Forn-qubit systems, our algorithm, which employs the easily prepared initial state $$\left\vert {0}^{n}\right\rangle \,\left\langle {0}^{n}\right\vert$$0n0n assumed to be noiseless, efficiently estimatesk-RDMs with $$\widetilde{{{{\mathcal{O}}}}}(k{n}^{k})$$Õ(knk) state copies and $$\widetilde{{{{\mathcal{O}}}}}(\sqrt{n})$$Õ(n) calibration measurements for GTM noise with constant fidelities. We show that our algorithm is robust against noise types like depolarizing, damping, andX-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.

A controlled-not (cnot) circuit is the key gadget for entangling qubits in quantum computing systems. However, the qubit connectivity of noisy intermediate-scale quantum (NISQ) devices is constrained by their limited-connectivity architecture. To improve the performance of cnot circuits on NISQ devices, we investigate the optimization of the size/depth of cnot circuits under the limited-connectivity architecture. We present a method that can optimize the size of any $n$-qubit cnot circuit $O\left(\frac{n^2}{log\delta }\right)$ on any connected graph with minimum degree $\delta$, and prove this bound is optimal for the regular graph. For the near-term sparsely connected structure, we additionally present a method that can optimize the size of any $n$-qubit cnot circuit to below $2n^2$. The numerical experiment shows that our method performs better than state-of-the-art results. Specifically, we present an example to illustrate the applicability of our algorithm. For the grid structure, which is commonly used in current quantum devices, we demonstrate that the depth of any $n$-qubit cnot circuit can be optimized to be linear in $n$ with certain ancillary qubits (ancillas). Experimental results indicate that this method has significant improvements compared with all of the existing methods. We additionally test our algorithms on the five-qubit IBMQ devices, and the experiments show that the measurement results of the optimized circuit with our algorithm are more robust to noise compared with the IBM mapping method.

In this paper, we study the local unitary (LU) classification for pairs (triples) of generalized Bell states, based on the local unitary equivalence of two sets. In detail, we first introduce some general unitary operators which give us more local unitary equivalent sets besides Clifford operators. And then we present two necessary conditions for local unitary equivalent sets which can be used to examine the local inequivalence. Following this approach, we completely classify all pairs in the $d\otimes d$ quantum system into ${\prod }_{j=1}^n(k_j+1)-1$ LU-inequivalent pairs when the prime factorization of $d={\prod }_{j=1}^n{p}_j^{k_j}$. Moreover, all triples in the $p^{\alpha }\otimes p^{\alpha }$ quantum system for prime $p$ can be partitioned into $\frac{(\alpha +3)}{6}{p}^{\alpha }+O\left(\alpha p^{\alpha -1}\right)$ LU-inequivalent triples; especially, when $\alpha =2$ and $p>2$, there are exactly $\lfloor \frac{5}{6}{p}^2\rfloor +\lfloor \frac{p-2}{6}+{(-1)}^{\lfloor \frac{p}{3}\rfloor }\frac{p}{3}\rfloor +3$ LU-inequivalent triples.

The solving of linear systems provides a rich area to investigate the use of nearer-term, noisy, intermediate-scale quantum computers. In this work, we discuss hybrid quantum-classical algorithms for heavily skewed linear systems for overdetermined and underdetermined cases. Our input model is such that the columns or rows of the matrix defining the linear system are given via quantum circuits of polylogarithmic depth and the number of circuits is much smaller than their Hilbert space dimension. Our algorithms have polylogarithmic dependence on the dimension and polynomial dependence in other natural quantities. In addition, we present an algorithm for the special case of a factorized linear system with run time polylogarithmic in the respective dimensions. At the core of these algorithms is the Hadamard test and in the second part of this paper, we consider the optimization of the circuit depth of this test. Given an $n$-qubit and $d$-depth quantum circuit $\mathcal{C}$, we can approximate $\langle 0|\mathcal{C}|0\rangle$ using $(n+s)$ qubits and $O\left({log}_{2}s+d{log}_2(n/s)+d\right)$-depth quantum circuits, where $s\le n$. In comparison, the standard implementation requires $n+1$ qubits and $O\left(dn\right)$ depth. Lattice geometries underlie recent quantum supremacy experiments with superconducting devices. We also optimize the Hadamard test for an $(l_1\times l_2)$ lattice with $l_1\times l_2=n$ and can approximate $\langle 0|\mathcal{C}|0\rangle$ with $(n+1)$ qubits and $O\left(d\left(l_1+l_2\right)\right)$-depth circuits. In comparison, the standard depth is $O\left(d{n}^2\right)$ in this setting. Both of our optimization methods are asymptotically tight in the case of one-depth quantum circuits $\mathcal{C}$.

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.