Quantum Information

I will report on an ongoing project to work out and exploit an analogue of Schur-Weyl duality for the Clifford group. Schur-Weyl establishes a one-one correspondence between irreps of the unitary group and those of the symmetric group. A similar program can be carried out for Cliffords. The permutations are then replaced by certain discrete orthogonal maps. As is the case for Schur-Weyl, this duality has many applications for problems in quantum information. It can be used, e.g., to derive quantum property tests for stabilizerness and Cliffordness, a new direct interpretation of the sum-negativity of Wigner functions, bounds on stabilizer rank, the construction of designs using few non-Clifford resources, etc. [arXiv:1609.08172, arXiv:1712.08628, arXiv:1906.07230, arXiv:out.soon].

Affleck, Kennedy, Lieb, and Tasaki (AKLT) constructed one-dimensional and two-dimensional spin models invariant under spin rotation. These are recognized as paradigmatic examples of symmetry-protected topological phases, including the spin-1 AKLT chain with a provable nonzero spectral gap that strongly supports Haldane’s conjecture on the spectral gap of integer chains. These states were shown to provide universal resource for quantum computation, in the framework of the measurement-based approach, including the spin-3/2 AKLT state on the honeycomb lattice and the spin-2 one on the square lattice, both of which display exponential decay in the correlation functions. However, the nonzero spectral in these 2D models had not been proved analytically for over 30 years, until very recently. I will review briefly our understanding of the quantum computational universality in the AKLT family. Then I will focus on demonstrating the nonzero spectral gap for several 2D AKLT models, including decorated honeycomb and decorated square lattices, and the undecorated degree-3 Archimedean lattices. In brief, we now have universal resource states that are ground states of provable gapped local Hamiltonians. Such a feature may be useful in creating the resource states by cooling the system and might further help the exploration into the quantum computational phases in generalized AKLT-Haldane phases.

The manipulation of quantum "resources" such as entanglement and coherence lies at the heart of quantum advantages and technologies. In practice, a particularly important kind of manipulation is to "purify" the quantum resources, since they are inevitably contaminated by noises and thus often lost their power or become unreliable for direct usage. Here we derive fundamental limitations on how effectively generic noisy resources can be purified enforced by the laws of quantum mechanics, which universally apply to any reasonable kind of quantum resource. Remarkably, it is impossible to achieve perfect resource purification, even probabilistically. Our theorems indicate strong limits on the efficiency of distillation, a widely-used type of resource purification routine that underpins many key applications of quantum information science. In particular, we present explicit lower bounds on the resource cost of magic state distillation, a leading scheme for realizing scalable fault-tolerant quantum computation

There is a standard generalization of stabilizer codes to work with qudits which have prime dimension, and a slightly less standard generalization for qudits whose dimension is a prime power. However, for prime power dimensions, the usual generalization effectively treats the qudit as multiple prime-dimensional qudits instead of one larger object. There is a finite field GF(q) with size equal to any prime power, and it makes sense to label the qudit basis states with elements of the finite field, but the usual stabilizer codes do not make use of the structure of the finite field. I introduce the true GF(q) stabilizer codes, a subset of the usual prime power stabilizer codes which do make full use of the finite field structure. The true GF(q) stabilizer codes have nicer properties than the usual stabilizer codes over prime power qudits and work with a lifted Pauli group, which has some interesting mathematical aspects to it.

A self-correcting quantum memory can store and protect quantum information for a time that increases without bound in the system size, without the need for active error correction. Unfortunately, the landscape of Hamiltonians based on stabilizer (subspace) codes is heavily constrained by numerous no-go results and it is not known if they can exist in three dimensions or less. In this talk, we will discuss the role of symmetry in self-correcting memories. Firstly, we will demonstrate that codes given by 2D symmetry-enriched topological (SET) phases that appear naturally on the boundary of 3D symmetry-protected topological (SPT) phases can be self-correcting -- provided that they are protected by an appropriate subsystem symmetry. Secondly, we discuss the feasibility of self-correction in Hamiltonians based on subsystem codes, guided by the concept of emergent symmetries. We present ongoing work on a new exactly solvable candidate model in this direction based on the 3D gauge color code. The model is a non-commuting, frustrated lattice model which we prove to have an energy barrier to all bulk errors. Finding boundary conditions that encode logical qubits and retain the bulk energy barrier remains an open question.

We introduce the hemicubic codes, a family of quantum codes obtained by associating qubits with the p-faces of the n-cube (for n>p) and stabilizer constraints with faces of dimension (p±1). The quantum code obtained by identifying antipodal faces of the resulting complex encodes one logical qubit into N=2n−p−1(np) physical qubits and displays local testability with a soundness of Ω(log−2(N)) beating the current state-of-the-art of log−3(N) due to Hastings. We exploit this local testability to devise an efficient decoding algorithm that corrects arbitrary errors of size less than the minimum distance, up to polylog factors. We then extend this code family by considering the quotient of the n-cube by arbitrary linear classical codes of length n. We establish the parameters of these generalized hemicubic codes. Interestingly, if the soundness of the hemicubic code could be shown to be 1/log(N), similarly to the ordinary n-cube, then the generalized hemicubic codes could yield quantum locally testable codes of length not exceeding an exponential or even polynomial function of the code dimension. (joint work with Vivien Londe and Gilles Zémor)

It is known that several sub-universal quantum computing models cannot be classically simulated unless the polynomial-time hierarchy collapses. However, these results exclude only polynomial-time classical simulations. In this talk, based on fine-grained complexity conjectures, I show more ``fine-grained" quantum supremacy results that prohibit certain exponential-time classical simulations. I also show the stabilizer rank conjecture under fine-grained complexity conjectures.

Consider the task of estimating the expectation value of an n-qubit tensor product observable in the output state of a shallow quantum circuit. This task is a cornerstone of variational quantum algorithms for optimization, machine learning, and the simulation of quantum many-body systems. In this talk I will describe three special cases of this problem which are "easy" for classical computers. This is joint work with Sergey Bravyi and Ramis Movassagh.

I will review the stabiliser rank and associated pure state magic monotone, the extent, [Bravyi et. al 2019]. Then I will discuss several new magic monotones that can be regarded as a generalisation of the extent monotone to mixed states [Campbell et. al., in preparation]. My talk will outline several nice theorems we can prove about these monotones relate to each other and how they are related to the runtime of new classical simulation algorithms.

The variational quantum eigensolver (VQE) is the leading candidate for practical applications of Noisy Intermediate Scale Quantum (NISQ) devices. The method has been widely implemented on small NISQ machines in both superconducting and ion trap implementations. I will review progress to date and discuss two questions . Firstly, how quantum mechanical are small VQE demonstrations? We will analyze this question using strong measurement contextuality. Secondly, can VQE be implemented at the scale of devices capable of exhibiting quantum supremacy, around 50 qubits? I will discuss some recent techniques to reduce the number of measurements required, which again use the concept of contextuality.

We take a resource-theoretic approach to the problem of quantifying nonclassicality in Bell scenarios. The resources are conceptualized as probabilistic processes from the setting variables to the outcome variables which have a particular causal structure, namely, one wherein the wings are only connected by a common cause. The distinction between classical and nonclassical is then defined in terms of whether or not a classical causal model can explain the correlations. The relative nonclassicality of such resources is quantified by considering their interconvertibility relative to the set of operations that can be implemented using a classical common cause (which correspond to local operations and shared randomness). Among other results, we show that the information contained in the degrees of violation of facet-defining Bell inequalities is not sufficient for quantifying nonclassicality, even though it is sufficient for witnessing nonclassicality. In addition to providing new insights on Bell nonclassicality, our work sets the stage for quantifying nonclassicality in more general causal networks and thus also for a resource-theoretic account of nonclassicality in computational settings. (Joint work with Elie Wolfe, David Schmid, Ana Belen Sainz, and Ravi Kunjwal)