The Quantum Formalization Drive

An error-correcting code protects information by adding redundancy. A quantum code must also account for two kinds of errors, commonly called $X$ and $Z$ errors. Quantum low-density parity-check codes use checks that each touch only a bounded number of qubits, while each qubit participates in only a bounded number of checks. Such locality matters when the…

Simulating the time evolution of a quantum system is a standard task in quantum algorithms: given a Hamiltonian, the algorithm must reproduce the phase acquired by each of its energy eigenstates. The cost is often measured by how many times the algorithm queries an operator describing that system. Low and Chuang's 2016 preprint connects this task to quantum…

The Quantum Approximate Optimization Algorithm (QAOA) of Farhi, Goldstone and Gutmann (arXiv:1411.4028) is a family of shallow quantum circuits for combinatorial optimization. At its lowest level, $p = 1$, the circuit has two real parameters, and its output is a random bit string. The question is how many constraints that string satisfies, compared with the…

Quantum state tomography reconstructs an unknown $d$-dimensional quantum state from measurements on many copies of it. For $n$ qubits, $d = 2^n$, so the number of copies a tomography procedure needs grows exponentially in $n$, and lower bounds on that number decide how far experiments can scale. In many experiments the only measurements available are Pauli…

This mission develops Chapter 8 of Quantum algorithms for optimizers by Giacomo Nannicini. The chapter's selected results describe the mathematical guarantees below.

This mission develops Chapter 5 of An Introduction to Quantum Error Correction and Fault-Tolerant Quantum Computation by Daniel Gottesman. The chapter's selected results describe the mathematical guarantees below.

Quantum information often describes a system through a state shared among many small registers. One would like to replace the state on a few registers by a product of their individual states, because product states are easier to estimate and optimize over. The usual finite quantum de Finetti theorems obtain such approximations from permutation symmetry. A…

Many problems in quantum information theory ask for the maximum of a linear functional over pairs of quantum states, one on each of two systems, subject to linear constraints on each state. The best success probability of an approximate quantum error-correcting code of given size is of this form: the encoder and the decoder are quantum channels, their Choi…

In a 1999 Vienna doctoral thesis, Gerhard Zauner conjectured that in every finite dimension d one can find d² unit vectors in complex d-space that are mutually as spread out as possible — any two sharing the same squared overlap 1/(d+1). Such a configuration, a symmetric informationally complete positive operator-valued measure (SIC-POVM), is the optimal…

Line bundles on a manifold $M$ are classified by their Chern class in $H^2(M,\mathbb Z)$. One degree up, gerbes as sheaves of groupoids (Brylinski 1993) realize classes in $H^3(M,\mathbb Z)$ geometrically. Bundle gerbes, introduced by Murray (1996), do this with bundles rather than sheaves, and they appear in index theory and Hamiltonian quantization (Carey,…

A centered Gaussian process with stationary increments is determined by a spectral measure $\sigma$ on $\mathbb R$: its covariance is $\int\chi_t(u)\overline{\chi_s(u)}\,d\sigma(u)$ with $\chi_t(u)=(e^{itu}-1)/u$. Brownian motion corresponds to Lebesgue measure and fractional Brownian motion to a power density. When $\sigma$ has a density $m$, a stochastic calculus for…

A quantum computer whose qubits sit close together is often disturbed by noise that couples to all of them in the same way. This is collective decoherence. When the noise can rotate the joint spin of the register about any axis, it is strong collective decoherence. Information stored in the degeneracy of the joint spin is then untouched by the noise. This…

In this paper we study the classifying theory of principal bundles in the parametrized setting, motivated by recent interest in higher gauge theory. Using simplicial techniques, we construct a product-preserving classifying space functor for groups in the category of spaces over a fixed space B. Additionally, we prove that the fiberwise geometric…

We study the entanglement entropy in confining theories with gravity duals using the holographic prescription of Ryu and Takayanagi. The entanglement entropy between a region and its complement is proportional to the minimal area of a bulk hypersurface ending on their border. We consider a disk in 2+1 dimensions and a ball in 3+1 dimensions and find in both…

We describe the local structure of an irreducible algebraic monoid $M$ at an idempotent element $e$. When $e$ is minimal, we show that $M$ is an induced variety over the kernel $MeM$ (a homogeneous space) with fibre the two-sided stabilizer $M_e$ (a connected affine monoid having a zero element and a dense unit group). This yields the irreducibility of stabilizers and…

In this paper we study a general family of multivariable Gaussian stochastic processes. Each process is prescribed by a fixed Borel measure $σ$ on $\mathbb R^n$. The case when $σ$ is assumed absolutely continuous with respect to Lebesgue measure was studied earlier in the literature, when $n=1$. Our focus here is on showing how different equivalence classes…

In some bicategories, the 1-cells are morphisms' between the 0-cells, such as functors between categories, but in others they are objects' over the 0-cells, such as bimodules, spans, distributors, or parametrized spectra. Many bicategorical notions do not work well in these cases, because the …

We prove that the mapping stack Map(Y,X) of topological stacks X and Y is again a topological stack if Y admits a compact groupoid presentation. If Y admits a locally compact groupoid presentation, we show that Map(Y,X) is a paratopological stack. In particular, it has a classifying space (hence, a natural weak homotopy type). We prove an invariance theorem…

Quantum networks will allow to implement communication tasks beyond the reach of their classical counterparts. A pressing and necessary issue for the design of quantum network protocols is the quantification of the rates at which these tasks can be performed. Here, we propose a simple recipe that yields efficiently computable lower and upper bounds on the…

Quantum Max-Cut asks for a high-energy state of an antiferromagnetic spin system specified by a weighted graph. Vertices carry qubits, and an edge rewards the two qubits for occupying their antisymmetric singlet state. A single qubit participates in several interactions, so maximizing each edge separately generally gives incompatible demands. The problem…

Quantum Max-Cut asks for a high-energy state of an antiferromagnetic spin system specified by a weighted graph. Vertices carry qubits, and an edge rewards the two qubits for occupying their antisymmetric singlet state. A single qubit participates in several interactions, so maximizing each edge separately generally gives incompatible demands. The problem…

The Fisher information of a real random variable measures how sharply its density is peaked: it is the information an observation $X=\theta+N$ carries about a translation parameter $\theta$, and by the Cramér–Rao bound its reciprocal is the smallest variance of an unbiased estimator of $\theta$. The Fisher information inequality (FII) says how this quantity…

Tensor networks represent the state of a many-body quantum system on $N$ sites as a contraction of small tensors, one per vertex of a graph. They underlie the matrix product state, PEPS and MERA methods of condensed-matter physics and quantum chemistry. Many physical systems have a global symmetry: a group $G$ acting by the same unitary $U_g$ on every site, under…

General relativity has a dynamical causal structure: whether one event can influence another depends on the metric, which is itself a physical variable. Quantum theory, in its standard formulations, presupposes a fixed causal structure: states evolve in a given time, and systems are composed in a given order. A theory of quantum gravity is expected to…

An extension of a Lie algebra $\mathfrak g$ by a Lie algebra $\mathfrak n$ is a Lie algebra $\widehat{\mathfrak g}$ containing $\mathfrak n$ as an ideal with quotient $\widehat{\mathfrak g}/\mathfrak n\cong\mathfrak g$. Extensions are how larger Lie algebras are assembled from smaller ones: central extensions produce the Virasoro and affine Kac–Moody algebras from…