Martin Kliesch

Dissipative quantum Church-Turing theorem.

We show that the time evolution of an open quantum system, described by a possibly time dependent Liouvillian, can be simulated by a unitary quantum circuit of a size scaling polynomially in the simulation time and the size of the system. An immediate consequence is that dissipative quantum computing is no more powerful than the unitary circuit model. Our result can be seen as a dissipative Church-Turing theorem, since it implies that under natural assumptions, such as weak coupling to an environment, the dynamics of an open quantum system can be simulated efficiently on a quantum computer. Formally, we introduce a Trotter decomposition for Liouvillian dynamics and give explicit error bounds. This constitutes a practical tool for numerical simulations, e.g., using matrix-product operators. We also demonstrate that most quantum states cannot be prepared efficiently.

Reliable quantum certification of photonic state preparations

Quantum technologies promise a variety of exciting applications. Even though impressive progress has been achieved recently, a major bottleneck currently is the lack of practical certification techniques. The challenge consists of ensuring that classically intractable quantum devices perform as expected. Here we present an experimentally friendly and reliable certification tool for photonic quantum technologies: an efficient certification test for experimental preparations of multimode pure Gaussian states, pure non-Gaussian states generated by linear-optical circuits with Fock-basis states of constant boson number as inputs, and pure states generated from the latter class by post-selecting with Fock-basis measurements on ancillary modes. Only classical computing capabilities and homodyne or hetorodyne detection are required. Minimal assumptions are made on the noise or experimental capabilities of the preparation. The method constitutes a step forward in many-body quantum certification, which is ultimately about testing quantum mechanics at large scales.

Positive Tensor Network Approach for Simulating Open Quantum Many-Body Systems.

Open quantum many-body systems play an important role in quantum optics and condensed matter physics, and capture phenomena like transport, the interplay between Hamiltonian and incoherent dynamics, and topological order generated by dissipation. We introduce a versatile and practical method to numerically simulate one-dimensional open quantum many-body dynamics using tensor networks. It is based on representing mixed quantum states in a locally purified form, which guarantees that positivity is preserved at all times. Moreover, the approximation error is controlled with respect to the trace norm. Hence, this scheme overcomes various obstacles of the known numerical open-system evolution schemes. To exemplify the functioning of the approach, we study both stationary states and transient dissipative behavior, for various open quantum systems ranging from few to many bodies.

Matrix-Product Operators and States: NP-Hardness and Undecidability

Tensor network states constitute an important variational set of quantum states for numerical studies of strongly correlated systems in condensed-matter physics, as well as in mathematical physics. This is specifically true for finitely correlated states or matrix-product operators, designed to capture mixed states of one-dimensional quantum systems. It is a well-known open problem to find an efficient algorithm that decides whether a given matrix-product operator actually represents a physical state that in particular has no negative eigenvalues. We address and answer this question by showing that the problem is provably undecidable in the thermodynamic limit and that the bounded version of the problem is NP-hard (nondeterministic-polynomial-time hard) in the system size. Furthermore, we discuss numerous connections between tensor network methods and (seemingly) different concepts treated before in the literature, such as hidden Markov models and tensor trains.

Direct certification of a class of quantum simulations

One of the main challenges in the field of quantum simulation and computation is to identify ways to certify the correct functioning of a device when a classical efficient simulation is not available. Important cases are situations in which one cannot classically calculate local expectation values of state preparations efficiently. In this work, we develop weak-membership formulations of the certification of ground state preparations. We provide a non-interactive protocol for certifying ground states of frustration-free Hamiltonians based on simple energy measurements of local Hamiltonian terms. This certification protocol can be applied to classically intractable analog quantum simulations: For example, using Feynman-Kitaev Hamiltonians, one can encode universal quantum computation in such ground states. Moreover, our certification protocol is applicable to ground states encodings of IQP circuits demonstration of quantum supremacy. These can be certified efficiently when the error is polynomially bounded.

Lieb-Robinson Bounds and the Simulation of Time-Evolution of Local Observables in Lattice Systems

This is an introductory text reviewing Lieb-Robinson bounds for open and closed quantum many-body systems. We introduce the Heisenberg picture for time-dependent local Liouvillians and state a Lieb-Robinson bound that gives rise to a maximum speed of propagation of correlations in many body systems of locally interacting spins and fermions. Finally, we discuss a number of important consequences concerning the simulation of time evolution and properties of ground states and stationary states.

Real-space renormalization yields finite correlations.

Real-space renormalization approaches for quantum lattice systems generate certain hierarchical classes of states that are subsumed by the multiscale entanglement renormalization Ansatz (MERA). It is shown that, with the exception of one spatial dimension, MERA states are actually states with finite correlations, i.e., projected entangled pair states (PEPS) with a bond dimension independent of the system size. Hence, real-space renormalization generates states which can be encoded with local effective degrees of freedom, and MERA states form an efficiently contractible class of PEPS that obey the area law for the entanglement entropy. It is further pointed out that there exist other efficiently contractible schemes violating the area law.

Mixing Properties of Stochastic Quantum Hamiltonians

Random quantum processes play a central role both in the study of fundamental mixing processes in quantum mechanics related to equilibration, thermalisation and fast scrambling by black holes, as well as in quantum process design and quantum information theory. In this work, we present a framework describing the mixing properties of continuous-time unitary evolutions originating from local Hamiltonians having time-fluctuating terms, reflecting a Brownian motion on the unitary group. The induced stochastic time evolution is shown to converge to a unitary design. As a first main result, we present bounds to the mixing time. By developing tools in representation theory, we analytically derive an expression for a local k-th moment operator that is entirely independent of k, giving rise to approximate unitary k-designs and quantum tensor product expanders. As a second main result, we introduce tools for proving bounds on the rate of decoupling from an environment with random quantum processes. By tying the mathematical description closely with the more established one of random quantum circuits, we present a unified picture for analysing local random quantum and classes of Markovian dissipative processes, for which we also discuss applications.

On the distribution of a product of N Gaussian random variables

The product of Gaussian random variables appears naturally in many applications in probability theory and statistics. It has been known that the distribution of a product of N such variables can be expressed in terms of a Meijer G-function. Here, we compute a similar representation for the corresponding cumulative distribution function (CDF) and provide a power-log series expansion of the CDF based on the theory of the more general Fox H-functions. Numerical computations show that for small values of the argument the CDF of products of Gaussians is well approximated by the lowest orders of this expansion. Analogous results are also shown for the absolute value as well as the square of such products of N Gaussian random variables. For the latter two settings, we also compute the moment generating functions in terms of Meijer G-functions.

Note on the saturation of the norm inequalities between diamond and nuclear norm.

The diamond norm plays an important role in quantum information and operator theory. Recently, it has also been proposed as a regularizer for low-rank matrix recovery. The norm constants that bound the diamond norm in terms of the nuclear norm (also known as trace norm) are explicitly known. This paper provides a simple characterization of all operators saturating the upper and lower bounds.The diamond norm plays an important role in quantum information and operator theory. Recently, it has also been proposed as a reguralizer for low-rank matrix recovery. The norm constants that bound the diamond norm in terms of the nuclear norm (also known as trace norm) are explicitly known. This note provides a simple characterization of all operators saturating the upper and the lower bound.