We provide an alternative proof of Wallman's [Quantum 2, 47 (2018)] and

Proctor's [Phys. Rev. Lett. 119, 130502 (2017)] bounds on the effect of

gate-dependent noise on randomized benchmarking (RB). Our primary insight is

that a RB sequence is a convolution amenable to Fourier space analysis, and we

adopt the mathematical framework of Fourier transforms of matrix-valued

functions on groups established in recent work from Gowers and Hatami [Sbornik:

Mathematics 208, 1784 (2017)]. We show explicitly that as long as our faulty

# All

Quantum gates (unitary gates) on physical systems are usually implemented by

controlling the Hamiltonian dynamics. When full descriptions of the

Hamiltonians parameters is available, the set of implementable quantum gates is

easily characterised by quantum control theory. In many real systems, however,

the Hamiltonians may include unknown parameters due to the difficulty of

precise measurements or instability of the system. In this paper, we consider

the situation that some parameters of the Hamiltonian are unknown, but we still

We investigate the uncertainty relation for the position of one electron in a

uniform magnetic field in the framework of the quantum estimation theory. Two

kinds of momenta, canonical one and mechanical one, are used to generate a

shift in the position of the electron. We first consider pure state models

whose wave function is in the ground state with zero angular momentum. The

model generated by the two-commuting canonical momenta becomes the

quasi-classical model, in which the symmetric logarithmic derivative quantum

We have developed a software library that simulates noisy quantum logic

circuits. We represent quantum states by their density matrices, and

incorporate possible errors in initialisation, logic gates, memory and

measurement using simple models. Our quantum simulator is implemented as a new

backend on IBM's open-source Qiskit platform. In this document, we provide its

description, and illustrate it with some simple examples.

We study amorphous systems with completely random sites and find that,

through constructing and exploring a concrete model Hamiltonian, such a system

can host an exotic phase of topological amorphous metal in three dimensions. In

contrast to the traditional Weyl semimetals, topological amorphous metals break

translational symmetry, and thus cannot be characterized by the first Chern

number defined based on the momentum space band structures. Instead, their

topological properties will manifest in the Bott index and the Hall

The study and prediction of chemical reactivity is one of the most important

application areas of molecular quantum chemistry. Large-scale, fully

error-tolerant quantum computers could provide exact or near-exact solutions to

the underlying electronic structure problem with exponentially less effort than

a classical computer thus enabling highly accurate predictions for comparably

large molecular systems. In the nearer future, however, only "noisy" devices

with a limited number of qubits that are subject to decoherence will be

The parity-preserving $U_A(1)\times U_a(1)$ massive QED$_3$ is ultraviolet

finiteness -- exhibits vanishing $\beta$-functions, associated to the gauge

coupling constants (electric and chiral charges) and the Chern-Simons mass

parameter, and all the anomalous dimensions of the fields -- as well as is

parity and gauge anomaly free at all orders in perturbation theory. The proof

is independent of any regularization scheme and it is based on the quantum

action principle in combination with general theorems of perturbative quantum

We consider the problem of maximizing a homogeneous polynomial on the unit

sphere and its hierarchy of Sum-of-Squares (SOS) relaxations. Exploiting the

polynomial kernel technique, we obtain a quadratic improvement of the known

convergence rate by Reznick and Doherty & Wehner. Specifically, we show that

the rate of convergence is no worse than $O(d^2/\ell^2)$ in the regime $\ell

\geq \Omega(d)$ where $\ell$ is the level of the hierarchy and $d$ the

dimension, solving a problem left open in the recent paper by de Klerk &

Implementing high-fidelity quantum control and reducing the effect of the

coupling between a quantum system and its environment is a major challenge in

developing quantum information technologies. Here, we show that there exists a

geometrical structure hidden within the time-dependent Schr\"odinger equation

that provides a simple way to view the entire solution space of pulses that

suppress noise errors in a system's evolution. In this framework, any

single-qubit gate that is robust against quasistatic noise to first order

In order to elucidate the role of spontaneous symmetry breaking in condensed

matter systems, we explicitly construct the ground state wave function for a

nonrelativistic theory of a two-fluid system of bosons. This can model either

superconductivity or superfluidity, depending on whether we assign a charge to

the particles or not. Since each nonrelativistic field $\Psi_j$ ($j=1,2$)

carries a phase $\theta_j$ and the Lagrangian is formally invariant under

shifts $\theta_j\to\theta_j+\alpha_j$ for independent $\alpha_j$, one can