Benchmarking quantum processors with quantum error correction
Almost everything a fault-tolerant quantum computer will do will be dedication to a single task: error correction. Performing computations will essentially be a side effect of the error correction process. To determine the performance of such a device, we can therefore focus on how well it implements the detection and correction of errors. In this talk I will present results from a minimal implementation of the repetition code on the 16 qubitRueschlikon device. I will also propose tests based on the surface code that could be applied to the emerging generation of quantum processors.
Qubit losses in topological quantum codes
Quantum information is the generalisation of the classical information theory to quantum systems. If for a classical computer the bits are the fundamental units, quantum computers are built on quantum bits currently implemented in different physical systems (atoms, ions, molecules, Josephson junctions). However, these setups are so delicate that qubits can be corrupted by environmental noise and even be completely lost from the apparatus resulting in the partial or total loss of the information there memorised. In this talk, we will consider how losses can affect a particular class of quantum codes (the so-called color codes). We introduce a protocol for dealing with losses and we show that checking whether the logical information is still recoverable is equivalent to a generalised percolation process. We numerically compute the associated qubit loss thresholds and show that color codes are robust against qubit loss.
Classical methods for quantum error correction
Quantum error correction protocols are an essential element in the design of any circuit-model quantum computer. In this work we introduce the coherent parity check (CPC) framework for quantum error correction. CPC codes have a fundamental structure in which quantum parity check measurements are stored coherently and compared over time. The specific advantage of the CPC code structure is that it provides a way of creating new stabilizer codes from the starting point of any sequence of parity checks. We show that this freedom in the choice of parity checks can be used to derive methods for the construction of distance-three quantum codes based on almost any distance-three classical code. Another feature of CPC codes is that they can be represented as factor graphs of the type commonly seen in classical error correction and machine learning. We outline a procedure for this mapping, and demonstrate how a quantum code can be derived by manipulating its factor graph representation. The aim of this factor graph mapping for CPC codes is to make it easier to adapt well-developed techniques from classical information theory for use with quantum codes.
Parallelized quantum error correction with fracton topological codes
Topological phases with fractal excitations have a large number of emergent symmetries that enforce a rigid structure on the excited states of the system. Remarkably, we find that the symmetries of a quantum error-correcting code based on a fractal phase enable us to design highly parallelized decoding algorithms. Here we propose decoding algorithms for the three-dimensional X-cube model where decoding is subdivided into a series of two-dimensional matching problems, thus simplifying the most time consuming component of the decoder significantly. In addition to this, we also show that we can leverage the rigid structure of the syndrome data to improve threshold error rates beyond those of other three-dimensional models with point-like excitations. With the explicit example we have considered here we conclude by motivating the exploration of parallelizable codes. In particular we discuss their relationship with other fracton topological codes together with other modern notions that have arisen in the field of quantum error correction such as single-shot codes and self correction.
Generic Emergence of Objectivity of Observables in Infinite Dimensions
Phys. Rev. Lett. 121, 160401 (2018)
Quantum Darwinism posits that information becomes objective whenever multiple observers indirectly probe a quantum system by each measuring a fraction of the environment. It was recently shown that objectivity of observables emerges generically from the mathematical structure of quantum mechanics, whenever the system of interest has finite dimensions and the number of environment fragments is large [F. G. S. L. Brandão, M. Piani, and P. Horodecki, Nat. Commun. 6, 7908 (2015)]. Despite the importance of this result, it necessarily excludes many practical systems of interest that are infinite dimensional, including harmonic oscillators. Extending the study of quantum Darwinism to infinite dimensions is a nontrivial task: we tackle it here by using a modified diamond norm, suitable to quantify the distinguishability of channels in infinite dimensions. We prove two theorems that bound the emergence of objectivity, first for finite mean energy systems, and then for systems that can only be prepared in states with an exponential energy cutoff. We show that the latter class includes any bounded-energy subset of single-mode Gaussian states.
Quantum compilation and natural language processing in one picture
For well over a decade, we developed an entirely pictorial (and of course, formally rigorous) presentation of quantum theory , and it was recently shown that graphical reasoning by means of ZX-calculus can reproduce all equational reasoning in Hilbert space [2a, 2b]. In practical terms, it is currently for example being used as the core of quantum compilation , as it allows for easy translation between different computational models, allows for automation, and has outperformed any other method for circuit reduction. At the present, experiments are also being setup aimed at establishing the age at which children could effectively learn quantum theory in this manner. Meanwhile, the pictorial language has also been successful in the study of natural language  which induces new quantum algorithms , and we have started to apply it to model cognition, where we employ GPT-alike models . In a recent development we are able to assign meaning to entire texts, which then looks like a quantum circuit, that reduces to a ZX-like diagram.
We present the key ingredients of the pictorial language language as well as their interpretation across disciplines, and the applications mentioned above.
 BC & A. Kissinger (2017) Picturing Quantum Processes. A first course on quantum theory and diagrammatic reasoning. Cambridge University Press.
[2a] E. Jeandel, S. Perdrix & R. Vilmart (2017) A Complete Axiomatisation of the ZX-Calculus for Clifford+T Quantum Mechanics. arXiv:1705.11151
[2b] K. F. Ng & Q. Wang (2017) A universal completion of the ZX-calculus. arXiv:1706.09877
 S. Clark, BC, E. Grefenstette, S. Pulman & M. Sadrzadeh (2013) A quantum teleportation inspired algorithm produces sentence meaning from word meaning and grammatical structure. arXiv:1305.0556
 W. Zeng & BC (2016) Quantum Algorithms for Compositional Natural Language Processing. arXiv:1608.01406
 J. Bolt, BC, F. Genovese, M. Lewis, D. Marsden & R. Piedeleu (2017) Interacting Conceptual Spaces I : Grammatical Composition of Concepts. arXiv:1703.08314
Decomposing quantum operations: towards a quantum compiler
Every quantum gate can be decomposed into a sequence of single-qubit
gates and controlled-NOTs. In many implementations, single-qubit gates
are relatively ‘cheap’ to perform compared to C-NOTs (for instance,
being less susceptible to noise), and hence it is desirable to minimize
the number of C-NOT gates required to implement a circuit.
I will consider the task of constructing a generic isometry from m
qubits to n qubits, while trying to minimize the number of C-NOT gates
required. I will show a lower bound and then give an explicit gate
decomposition that gets within a factor of about two of this bound.
Through Stinespring’s theorem this points to a C-NOT-efficient way to
perform an arbitrary quantum operation. However, exploiting randomness
and quantum measurements leads to more efficient ways to build such
Solving Optimisation Problems with Quantum Walks
I will explain how to use quantum walks to solve optimisation problems
that are known to be amenable to adiabatic quantum computing or quantum
annealing. Examples of unstructured search and spin glasses will be
used to explain some of the factors that determine how well it works.
Constraints, disorder and localisation in Rydberg lattice gases
Rydberg lattice gases provide a versatile platform for studies of quantum few-body and many-body phenomena with applications ranging from quantum information processing to simulations of complex condensed matter systems. Of particular interest is the so-called facilitation mechanism (or anti-blockade), where the excitation of an atom to a Rydberg state is strongly enhanced in the vicinity of an already excited atom. This effect is of broad relevance and exploited in the design of quantum gates as well as in protocols for dissipative quantum state preparation. In the many-body context it effectuates an aggregation mechanism, where an initial Rydberg excitation seed triggers a dynamical growth of excitation clusters, and it enables the implementation of kinetic constraints thereby connecting to the physics of glass-forming substances.
I will focus in my talk on the dynamics of Rydberg excitations in an optical tweezer array under facilitation conditions . Due to the finite temperature the atomic positions are randomly spread, an effect that leads to quenched correlated disorder in the interatomic interaction strengths. This drastically affects the facilitation dynamics and may lead to Anderson localisation in Fock space [1,2]. Furthermore, the system features an unusual many-body localisation scenario in the sense that the it permits a mapping to fermions subject to non-local and disordered interactions. I will discuss the time-evolution of the entanglement entropy and population imbalance and will analyse the level-spacing statistics for this peculiar setting .
 M. Marcuzzi, J. Minar, D. Barredo, S. de Leseleuc, H. Labuhn, T. Lahaye, A. Browaeys, E. Levi and I. Lesanovsky, Facilitation dynamics and localization phenomena in Rydberg lattice gases with position disorder, Physical Review Letters 118, 063606 (2017)
 M. Ostmann, M. Marcuzzi, J. Minár and I. Lesanovsky, Synthetic lattices, flat bands and localization in Rydberg quantum simulators, arXiv:1802.00379, Quantum Science and Technology (2019)
 M. Ostmann, M. Marcuzzi, J.P. Garrahan and I. Lesanovsky, Localization in spin chains with facilitation constraints and disordered interactions, arXiv:1811.01667
Simulating unitary circuits of finite depth and infinite width with quantum channels
We introduce a numerical approach simulate unitary circuits of finite depth and infinite width. The unitary dynamics is encoded in a (sequence of) quantum channels acting on an ancilla space. The spectra of the reduced density matrices of a half-infinite region and the ancilla coincide, allowing for efficient evaluation of the entanglement spectrum and Rényi entropies. We benchmark our method on random unitary circuits, where many analytic results are available.
Simulating non-Markovian stochastic processes with quantum physics
Stochastic processes are as ubiquitous throughout the quantitative sciences as they are notorious for being difficult to simulate and predict. Weather patterns, stock prices, and biological evolution are just some of the most prominent examples.
In the last decades a sophisticated framework, called ‘computational mechanics’, has been developed that studies the complexity of such processes in terms of the minimal memory required for their simulation. More recently, it was discovered that this memory requirement for simulation may be further reduced by using a quantum instead of a classical memory substrate.
Based on these results, we have developed a generic method for constructing a unitary quantum simulators for a large class of stochastic processes. In this talk I will give a brief an introduction to computational mechanics and statistical complexity as well as their extension to quantum memory. I will then describe a proposal for the construction of a unitary quantum simulator which is applicable to a large class of stochastic processes. Finally, I will highlight some of the implications of the results — namely, unbounded scaling advantage, emergent ambiguity of process complexity when comparing quantum and classical models, and vanishing of time-reversal asymmetry in the quantum case.
Binder et al., PRL 120, 240502 (2018)
Undecidability of the Spectral Gap in One Dimension
The spectral gap problem – determining whether the energy spectrum of a system has an energy gap above ground state, or if there is a continuous range of low-energy excitations – pervades quantum many-body physics. Recently, this important problem was shown to be undecidable for quantum systems in two (or more) spatial dimensions: it is provably impossible to determine in general whether a system is gapped or gapless, a result which has many unexpected consequences for the physics of such systems. However, there are many indications that one dimensional systems are simpler than their higher-dimensional counterparts: for example, they cannot have thermal phase transitions or topological order, and there exist highly-effective numerical algorithms such as DMRG for gapped 1D systems, exploiting the fact that such systems obey an entropy area-law. Furthermore, the spectral gap undecidability construction crucially relied on aperiodic tilings, which are easily seen to be impossible in 1D. So does the spectral gap problem become decidable in 1D? In this paper we prove this is not the case, by constructing a family of 1D spin chains with translationally-invariant nearest neighbour interactions with undecidable spectral gap. This not only proves that the spectral gap of 1D systems is just as intractable, but also predicts the existence of qualitatively new types of complex physics in 1D spin chains. In particular, it implies there are 1D systems with constant spectral gap and unique classical ground state for all systems sizes up to an uncomputably large size, whereupon they switch to a gapless behaviour with dense spectrum.
A single-device, device-independent pseudo-secret random qubits
based on joint work with A. Cojocaru, L. Colisson and E. Kashefi arXiv:1802.08759
Abstract: We define the functionality of delegated pseudo-secret random qubit generator (PSRQG), where a classical client can instruct the preparation of a sequence of random qubits at some distant party. Their classical description is (computationally) unknown to any other party (including the distant party preparing them) but known to the client. We emphasize the unique feature that no quantum communication is required to implement PSRQG. Due to the fact that only the distant party has a quantum device, we can view such protocol as a single-device, device-independent protocol. The role of non-locality is replaced by the computational intractability of certain problems. We give a concrete protocol realising this functionality based on the hardness of the learning-with-errors problem. This enables classical clients to perform a class of quantum communication protocols with only a public classical channel with a quantum server. Important examples of such protocols include blind quantum computation, verifiable quantum computation and secure multiparty quantum computation.
Superpositions of mechanical processes and irreversibility
In Newtonian mechanics, any closed-system dynamics of a composite system in a pure state will leave all its individual subsystems in pure states, however this fails dramatically in quantum mechanics due to the existence of quantum entanglement. Here we introduce the notion of coherent superpositions of mechanical processes. This in turn leads to the notion of `decomposable’ and `non-decomposable’ quantum coherence and gives a new perspective on both recent analysis in the resource theory of asymmetry as well as early foundational analysis in the theory of classical random variables. Within the context of recent fluctuation relations, originally framed in terms of quantum channels, we show that coherent work processes play exactly the same role as their classical counterparts, and so provide a simple physical primitive for quantum coherence in such systems. We also introduce a pure state effective potential that is a flexible tool with which to analyze the coherent component of the fluctuation relations, leads to a notion of temperature-dependent mean coherence, provides connections with multi-partite entanglement, and gives a hierarchy of quantum corrections to the classical Crooks relation in powers of inverse temperature.