Quantum Information

When studying (definite or indefinite) causal orderings of processes, it is often useful to consider higher-order processes, i.e. processes which take other processes as their input. However, as a recent no-go result of Guerin et al indicates, our naive first-order notions of "composition" of processes become ill-defined at higher-order. Unlike state spaces, there are multiple non-equivalent notions of "joint system" for process spaces and many different ways one might attempt to plug processes together, with only some giving well-defined (i.e. normalised) processes as outputs. While this starts to look a bit like the Wild West, I'll show in this talk that we can get quite a bit of mileage from considering just two kinds of joint systems: a "non-signalling" tensor product, and a (de Morgan dual) "signalling" product. The interaction between these two products has in fact been well-understood by logicians since the 1980s in a very different disguise: multiplicative linear logic. Using this connection, I'll show how a set of "contractibility" criteria due to Danos and Regnier give a relatively simple, dimension-independent technique for determining whether an arbitrary plugging of higher-order processes is well-defined.

I will present an extension of the recent theory of quantum causal models to cyclic causal structures. This offers a novel causal perspective on processes beyond those corresponding to standard circuits, such as processes with dynamical causal order and causally nonseparable processes, including processes violating causal inequalities. As an application, I will use the algebraic structure of process operators that is induced by the causal structure to prove that all unitarily extendible bipartite processes are causally separable, i.e., their unitary extensions are variations of the quantum SWITCH. Remarkably, the latter implies that all unitarily extendible tripartite quantum processes have realizations on time-delocalized systems within standard quantum mechanics. This includes, in particular, classical processes violating causal inequalities, which admit simple implementations! I will explain what the violation of causal inequalities implies for the variables of interest in these implementations. The answer is given again by the theory of cyclic causal models. Based on joint works with Jonathan Barrett, Cyril Branciard, Robin Lorenz, and Julian Wechs.

The quantum SWITCH is the simplest example of indefinite causal structure. Technically, it is a higher-order transformation that takes two physical processes A and B in input and combines them in a coherent superposition of two alternative orders, AB and BA. In the past decade, the quantum SWITCH has been the object of active research, both theoretically and experimentally. In this talk, I will review the state of the art, and outline two new applications to quantum Shannon theory and quantum metrology.

The color code is a topological quantum code with many valuable fault-tolerant logical gates. Its two-dimensional version may soon be realized with currently available superconducting hardware despite constrained qubit connectivity. In the talk, I will focus on how to perform error correction with the color code in d ≥ 2 dimensions. I will describe an efficient color code decoder, the Restriction Decoder, which uses as a subroutine any toric code decoder. I will also present numerical estimates of the storage threshold of the Restriction Decoder for the triangular color code against circuit-level depolarizing noise. Based on arXiv:1905.07393 and arXiv:1911.00355.

In the AdS/CFT correspondence, global symmetries of the CFT are realized as local symmetries of AdS; this feature underlies the error-correcting property of AdS. I will explain how this allows AdS3 to realize multiple redundant computations of any CFT2 correlation function in the form of networks of Wilson lines. The main motivation is to rigorously define the CFT at a cutoff and study it as a model of computational complexity; in that regard we will find agreement with the holographic "Complexity = Volume" proposal. But the framework might be useful more generally.

In a continuum field theory the Hilbert space does not factorize into local tensor products. How then can we define entanglement and the basic protocols of quantum information theory? In this talk we will show how the factorization problem can be solved in a class of 2D conformal field theories by directly appealing to the fusion rules. The solution suggests a tensor network description of a CFT path integral using the OPE data.

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