Related papers: Efficient Unitary T-designs from Random Sums
A unitary state $t$-design is an ensemble of pure quantum states whose moments match up to the $t$-th order those of states uniformly sampled from a $d$-dimensional Hilbert space. Typically, unitary state $t$-designs are obtained by…
For a Haar random set $\mathcal{S}\subset U(d)$ of quantum gates we consider the uniform measure $\nu_\mathcal{S}$ whose support is given by $\mathcal{S}$. The measure $\nu_\mathcal{S}$ can be regarded as a…
Combinatorial $t$-designs have nice applications in coding theory, finite geometries and several engineering areas. There are two major methods of constructing $t$-designs. One of them is via group actions of certain permutation groups…
This paper presents a quantum algorithm for efficiently computing partial sums and specific weighted partial sums of quantum state amplitudes. Computation of partial sums has important applications, including numerical integration,…
Matchgate unitaries are ubiquitous in quantum computation due to their relation to non-interacting fermions and because they can be used to benchmark quantum computers. Implementing such unitaries on fault-tolerant devices requires first…
The purpose of this article is to study the eigenvalues $u_1^{\, t}=e^{it\theta_1},\dots,u_N^{\,t}=e^{it\theta_N}$ of $U^t$ where $U$ is a large $N\times N$ random unitary matrix and $t>0$. In particular we are interested in the typical…
Unitary $k$-designs are probabilistic ensembles of unitary matrices whose first $k$ statistical moments match that of the full unitary group endowed with the Haar measure. In prior work, we showed that the automorphism group of classical…
We propose a class of randomized quantum algorithms for the task of sampling from matrix functions, without the use of quantum block encodings or any other coherent oracle access to the matrix elements. As such, our use of qubits is purely…
Larger multi-qubit quantum gates allow shallower, more efficient quantum circuits, which could decrease the prohibitive effect of noise on algorithms for noisy intermediate-scale quantum (NISQ) devices and fault-tolerant error correction…
In this paper, we study the close relationship between Reed-Muller codes and single-qubit phase gates from the perspective of $T$-count optimization. We prove that minimizing the number of $T$ gates in an $n$-qubit quantum circuit over CNOT…
Unitary designs are widely used in quantum computation, but in many practical settings it suffices to construct a diagonal state design generated with unitary gates diagonal in the computational basis. In this work, we introduce a simple…
Quantum circuits of many qubits are extremely difficult to realize; thus, the number of qubits is an important metric in a quantum circuit design. Further, scalable and reliable quantum circuits are based on Clifford + T gates. An efficient…
We present a simple but general framework for constructing quantum circuits that implement the multiply-controlled unitary $\text{Select}(H) \equiv \sum_\ell |\ell\rangle\langle\ell|\otimes H_\ell$, where $H = \sum_\ell H_\ell$ is the…
We prove that random quantum circuits on any geometry, including a 1D line, can form approximate unitary designs over $n$ qubits in $\log n$ depth. In a similar manner, we construct pseudorandom unitaries (PRUs) in 1D circuits in…
In this paper, we first briefly review some recent results on the distribution of the maximal eigenvalue of a $(N\times N)$ random matrix drawn from Gaussian ensembles. Next we focus on the Gaussian Unitary Ensemble (GUE) and by suitably…
We have established the method of characterizing the unitary design generated by a symmetric local random circuit. Concretely, we have shown that the necessary and sufficient condition for the circuit asymptotically forming a t-design is…
Quantum algorithms claim significant speedup over their classical counterparts for solving many problems. An important aspect of many of these algorithms is the existence of a quantum oracle, which needs to be implemented efficiently in…
Recent work has shown that one can efficiently learn fermionic Gaussian unitaries, also commonly known as nearest-neighbor matchcircuits or non-interacting fermionic unitaries. However, one could ask a similar question about unitaries that…
Randomness is a fundamental resource in quantum information, with crucial applications in cryptography, algorithms, and error correction. A central challenge is to construct unitary $k$-designs that closely approximate Haar-random unitaries…
We study systematically numerical method into constructing a universal quantum gate set for topological quantum computation (TQC) using SU(2)k anyon models. The F-matrices and R-symbol were computed through the q-deformed representation…