Publication list

  20 Jun 2023

Decoding the Projective Transverse Field Ising Model

F. Roser, H. P. Büchler, N. Lang

arXiv:2303.03081 [quant-ph]

The competition between non-commuting projective measurements in discrete quantum circuits can give rise to entanglement transitions. It separates a regime where initially stored quantum information survives the time evolution from a regime where the measurements destroy the quantum information. Here we study one such system - the projective transverse field Ising model - with a focus on its capabilities as a quantum error correction code. The idea is to interpret one type of measurement as an error and the other type as a syndrome measurement. We demonstrate that there is a finite threshold below which quantum information encoded in an initially entangled state can be retrieved reliably. In particular, we implement the maximum likelihood decoder to demonstrate that the error correction threshold is distinct from the entanglement transition. This implies that there is a finite regime where quantum information is protected by the projective dynamics, but cannot be retrieved by using syndrome measurements.

8 Jun 2023

Noise can be helpful for variational quantum algorithms

J. Liu, F. Wilde, A. A. Mele, L. Jiang and J. Eisert

arXiv:2210.06723 [quant-ph]

Saddle points constitute a crucial challenge for first-order gradient descent algorithms. In notions of classical machine learning, they are avoided for example by means of stochastic gradient descent methods. In this work, we provide evidence that the saddle points problem can be naturally avoided in variational quantum algorithms by exploiting the presence of stochasticity. We prove convergence guarantees of the approach and its practical functioning at hand of examples. We argue that the natural stochasticity of variational algorithms can be beneficial for avoiding strict saddle points, i.e., those saddle points with at least one negative Hessian eigenvalue. This insight that some noise levels could help in this perspective is expected to add a new perspective to notions of near-term variational quantum algorithms.

8 May 2023

Local integrals of motion and the stability of many-body localisation in disorder-free systems

C. Bertoni, J. Eisert, A. Kshetrimayum, A. Nietner and S. J. Thomson

arXiv:2208.14432 [cond-mat.dis-nn]

Many-body localisation in disordered systems in one spatial dimension is typically understood in terms of the existence of an extensive number of (quasi)-local integrals of motion (LIOMs) which are thought to decay exponentially with distance and interact only weakly with one another. By contrast, little is known about the form of the integrals of motion in disorder-free systems which exhibit localisation. Here, we explicitly compute the LIOMs for disorder-free localised systems, focusing on the case of a linearly increasing potential. We show that while in the absence of interactions, the LIOMs decay faster than exponentially, the addition of interactions leads to the formation of a spatially extended plateau. We study how varying the linear slope affects the localisation properties of the LIOMs, finding that there is a significant finite-size dependence, and present evidence that adding a weak harmonic potential does not result in typical many-body localisation phenomenology. By contrast, the addition of disorder has a qualitatively different effect, dramatically modifying the properties of the LIOMs.

  11 Apr 2023

Shallow shadows: Expectation estimation using low-depth random Clifford circuits

C. Bertoni, J. Haferkamp, M. Hinsche, M. Ioannou, J. Eisert and H. Pashayan

arXiv:2209.12924 [quant-ph]

We provide practical and powerful schemes for learning many properties of an unknown n-qubit quantum state using a sparing number of copies of the state. Specifically, we present a depth-modulated randomized measurement scheme that interpolates between two known classical shadows schemes based on random Pauli measurements and random Clifford measurements. These can be seen within our scheme as the special cases of zero and infinite depth, respectively. We focus on the regime where depth scales logarithmically in n and provide evidence that this retains the desirable properties of both extremal schemes whilst, in contrast to the random Clifford scheme, also being experimentally feasible. We present methods for two key tasks; estimating expectation values of certain observables from generated classical shadows and, computing upper bounds on the depth-modulated shadow norm, thus providing rigorous guarantees on the accuracy of the output estimates. We consider observables that can be written as a linear combination of poly(n) Paulis and observables that can be written as a low bond dimension matrix product operator. For the former class of observables both tasks are solved efficiently in n. For the latter class, we do not guarantee efficiency but present a method that works in practice; by variationally computing a heralded approximate inverses of a tensor network that can then be used for efficiently executing both these tasks.

  10 Mar 2023

Computational advantage of quantum random sampling

D. Hangleiter and J. Eisert

arXiv:2206.04079 [quant-ph]

Quantum random sampling is the leading proposal for demonstrating a computational advantage of quantum computers over classical computers. Recently, first large-scale implementations of quantum random sampling have arguably surpassed the boundary of what can be simulated on existing classical hardware. In this article, we comprehensively review the theoretical underpinning of quantum random sampling in terms of computational complexity and verifiability, as well as the practical aspects of its experimental implementation using superconducting and photonic devices and its classical simulation. We discuss in detail open questions in the field and provide perspectives for the road ahead, including potential applications of quantum random sampling.

22 Jan 2023

A note on lower bounds to variational problems with guarantees

J. Eisert

arXiv:2301.06142 [quant-ph]

Variational methods play an important role in the study of quantum many body problems, both in the flavour of classical variational principles based on tensor networks as well as of quantum variational principles in near-term quantum computing. This brief pedagogical note stresses that for translationally invariant lattice Hamiltonians, one can easily derive efficiently computable lower bounds to ground state energies that can and should be compared with variational principles providing upper bounds. As small technical results, it is shown that (i) the Anderson bound and a (ii) common hierarchy of semi-definite relaxations both provide approximations with performance guarantees that scale like a constant in the energy density for cubic lattices. (iii) Also, the Anderson bound is systematically improved as a hierarchy of semi-definite relaxations inspired by the marginal problem.

26 Feb 2023

Exponentially tighter bounds on limitations of quantum error mitigation

Y. Quek, D. Stilck França, S. Khatri, J. Jakob Meyer and J. Eisert

arXiv:2210.11505 [quant-ph]

Quantum error mitigation has been proposed as a means to combat unwanted and unavoidable errors in near-term quantum computing using no or few additional quantum resources, in contrast to fault-tolerant schemes that come along with heavy overheads. Error mitigation has been successfully applied to reduce noise in near-term applications. In this work, however, we identify strong limitations to the degree to which quantum noise can be effectively `undone' for larger system sizes. We set up a framework that rigorously captures large classes of error mitigation schemes in use today. The core of our argument combines fundamental limits of statistical inference with a construction of families of random circuits that are highly sensitive to noise. We show that even at poly log log n depth, a super-polynomial number of samples is needed in the worst case to estimate the expectation values of noiseless observables, the principal task of error mitigation. Notably, our construction implies that scrambling due to noise can kick in at exponentially smaller depths than previously thought. They also impact other near-term applications, constraining kernel estimation in quantum machine learning, causing an earlier emergence of noise-induced barren plateaus in variational quantum algorithms and ruling out exponential quantum speed-ups in estimating expectation values in the presence of noise or preparing the ground state of a Hamiltonian.

4 Jan 2023

Measuring out quasi-local integrals of motion from entanglement

B. Lu, C. Bertoni, S. J. Thomson, J. Eisert


Quasi-local integrals of motion are a key concept underpinning the modern understanding of many-body localisation, an intriguing phenomenon in which interactions and disorder come together. Despite the existence of several numerical ways to compute them - and astoundingly in the light of the observation that much of the phenomenology of many properties can be derived from them - it is not obvious how to directly measure aspects of them in real quantum simulations; in fact, the smoking gun of their experimental observation is arguably still missing. In this work, we propose a way to extract the real-space properties of such quasi-local integrals of motion based on a spatially-resolved entanglement probe able to distinguish Anderson from many-body localisation from non-equilibrium dynamics. We complement these findings with a new rigorous entanglement bound and compute the relevant quantities using tensor networks. We demonstrate that the entanglement gives rise to a well-defined length scale that can be measured in experiments.

 4 Dec 2022

Tensor network study of the spin-1/2 Heisenberg anti-ferromagnet on the Shuriken lattice

P. Schmoll, A. Kshetrimayum, J. Naumann, J. Eisert and Y. Iqbal

arXiv:2211.16932 [cond-mat.str-el]

We investigate the ground state of the spin S=1/2 Heisenberg anti-ferromagnet on the Shuriken lattice, also in the presence of an external magnetic field. To this end, we employ two-dimensional tensor network techniques based on infinite projected entangled pair and simplex states considering states with different sizes of the unit cells. We show that a valence bond crystal with resonances over length six loops emerges as the ground state (at any given finite bond dimension) yielding the lowest reported estimate of the ground state energy E0/J=−0.4410±0.0001 for this model, estimated in the thermodynamic limit. We also study the model in the presence of an external magnetic field and find the emergence of 0, 1/3 and 2/3 magnetization plateaus with states respecting translation and point group symmetries that feature loop-four plaquette resonances instead.

2 Nov 2022

Computational advantage of quantum random sampling. arXiv:2206.04079

D. Hangleiter and J. Eisert

arXiv:2206.04079 [quant-ph]

Quantum random sampling is the leading proposal for demonstrating a computational advantage of quantum computers over classical computers. Recently, first large-scale implementations of quantum random sampling have arguably surpassed the boundary of what can be simulated on existing classical hardware. In this article, we comprehensively review the theoretical underpinning of quantum random sampling in terms of computational complexity and verifiability, as well as the practical aspects of its experimental implementation using superconducting and photonic devices and its classical simulation. We discuss in detail open questions in the field and provide perspectives for the road ahead, including potential applications of quantum random sampling.

 31 Oct 2022

Finite temperature tensor network algorithm for frustrated two-dimensional quantum materials

P. Schmoll, C. Balz, B. Lake, J. Eisert and A. Kshetrimayum

arXiv:2211.00121 [cond-mat.str-el]

Aimed at a more realistic classical description of natural quantum systems, we present a two-dimensional tensor network algorithm to study finite temperature properties of frustrated model quantum systems and real quantum materials. For this purpose, we introduce the infinite projected entangled simplex operator ansatz to study thermodynamic properties. To obtain state-of-the-art benchmarking results, we explore the highly challenging spin-1/2 Heisenberg anti-ferromagnet on the Kagome lattice, a system for which we investigate the melting of the magnetization plateaus at finite magnetic field and temperature. Making close connection to actual experimental data of real quantum materials, we go on to studying the finite temperature properties of Ca10Cr7O28. We compare the magnetization curve of this material in the presence of an external magnetic field at finite temperature with classically simulated data. As a first theoretical tool that incorporates both thermal fluctuations as well as quantum correlations in the study of this material, our work contributes to settling the existing controversy between the experimental data and previous theoretical works on the magnetization process.

  17 Oct 2022

A compiler for universal photonic quantum computers

F. Zilk, K. Staudacher, T. Guggemos, K. Fürlinger, D. Kranzlmüller and P. Walther

arXiv:2210.09251 [quant-ph]

Photons are a natural resource in quantum information, and the last decade showed significant progress in high-quality single photon generation and detection. Furthermore, photonic qubits are easy to manipulate and do not require particularly strongly sealed environments, making them an appealing platform for quantum computing. With the one-way model, the vision of a universal and large-scale quantum computer based on photonics becomes feasible. In one-way computing, the input state is not an initial product state, but a so-called cluster state. A series of measurements on the cluster state's individual qubits and their temporal order, together with a feed-forward procedure, determine the quantum circuit to be executed. We propose a pipeline to convert a QASM circuit into a graph representation named measurement-graph (m-graph), that can be directly translated to hardware instructions on an optical one-way quantum computer. In addition, we optimize the graph using ZX-Calculus before evaluating the execution on an experimental discrete variable photonic platform.

9 Oct 2022

Estimating gate-set properties from random sequences

J. Helsen, M. Ioannou, I. Roth, J. Kitzinger, E. Onorati, A. H. Werner and J. Eisert

arXiv:2110.13178v2 [quant-ph]

With quantum computing devices increasing in scale and complexity, there is a growing need for tools that obtain precise diagnostic information about quantum operations. However, current quantum devices are only capable of short unstructured gate sequences followed by native measurements. We accept this limitation and turn it into a new paradigm for characterizing quantum gate-sets. A single experiment - random sequence estimation - solves a wealth of estimation problems, with all complexity moved to classical post-processing. We derive robust channel variants of shadow estimation with close-to-optimal performance guarantees and use these as a primitive for partial, compressive and full process tomography as well as the learning of Pauli noise. We discuss applications to the quantum gate engineering cycle, and propose novel methods for the optimization of quantum gates and diagnosing cross-talk.

 28 Sept 2022

Scalably learning quantum many-body Hamiltonians from dynamical data

F. Wilde, A. Kshetrimayum, I. Roth, D. Hangleiter, R. Sweke and J. Eisert

arXiv:2209.14328 [quant-ph]

The physics of a closed quantum mechanical system is governed by its Hamiltonian. However, in most practical situations, this Hamiltonian is not precisely known, and ultimately all there is are data obtained from measurements on the system. In this work, we introduce a highly scalable, data-driven approach to learning families of interacting many-body Hamiltonians from dynamical data, by bringing together techniques from gradient-based optimization from machine learning with efficient quantum state representations in terms of tensor networks. Our approach is highly practical, experimentally friendly, and intrinsically scalable to allow for system sizes of above 100 spins. In particular, we demonstrate on synthetic data that the algorithm works even if one is restricted to one simple initial state, a small number of single-qubit observables, and time evolution up to relatively short times. For the concrete example of the one-dimensional Heisenberg model our algorithm exhibits an error constant in the system size and scaling as the inverse square root of the size of the data set.

19 Sept 2022

Experimental observation of curved light-cones in a quantum field simulator

M. Tajik, M. Gluza, N. Sebe, P. Schüttelkopf, F. Cataldini, J. Sabino, F. Møller, S.-C. Ji, S. Erne, G. Guarnieri, S. Sotiriadis, J. Eisert and J. Schmiedmayer

arXiv:2209.09132 [cond-mat.quant-gas]

We investigate signal propagation in a quantum field simulator of the Klein-Gordon model realized by two strongly coupled parallel one-dimensional quasi-condensates. By measuring local phononic fields after a quench, we observe the propagation of correlations along sharp light-cone fronts. If the local atomic density is inhomogeneous, these propagation fronts are curved. For sharp edges, the propagation fronts are reflected at the system's boundaries. By extracting the space-dependent variation of the front velocity from the data, we find agreement with theoretical predictions based on curved geodesics of an inhomogeneous metric. This work extends the range of quantum simulations of non-equilibrium field dynamics in general spacetime metrics.

  Jul - Aug 2022

Accelerating HPC With Quantum Computing: It Is a Software Challenge Too

M. Schulz, M. Ruefenacht, D. Kranzlmüller and L. Schulz


With quantum computing (QC) maturing, high-performance computing (HPC) centers are already preparing to host early-phase production versions of such systems. Unlike their experimental predecessors in physics laboratories, with a very small and dedicated user community, this next generation of systems needs to serve a wider user community and must work in concert with existing HPC systems and software stacks. This article describes our vision for an integrated ecosystem that combines existing HPC and evolving quantum software stacks into a single system to enable a common and continuous user experience. This integration comes with several major challenges as quantum systems pose significantly different requirements including increased need for compilation at run time, long optimization times, statistical evaluations of results, and the need to work with few centralized resources. To overcome these challenges, new scheduling approaches on the HPC side and new programming approaches on the QC side are required.

26 Jul 2022

Entanglement estimation in tensor network states via sampling

N. Feldman, A. Kshetrimayum, J. Eisert and M. Goldstein

PRX Quantum 3, 030312 (2022)

We introduce a method for extracting meaningful entanglement measures of tensor network states in general dimensions. Current methods require the explicit reconstruction of the density matrix, which is highly demanding, or the contraction of replicas, which requires an effort exponential in the number of replicas and which is costly in terms of memory. In contrast, our method requires the stochastic sampling of matrix elements of the classically represented reduced states with respect to random states drawn from simple product probability measures constituting frames. Even though not corresponding to physical operations, such matrix elements are straightforward to calculate for tensor network states, and their moments provide the Rényi entropies and negativities as well as their symmetry-resolved components. We test our method on the one-dimensional critical XX chain and the two-dimensional toric code in a checkerboard geometry. Although the cost is exponential in the subsystem size, it is sufficiently moderate so that - in contrast with other approaches - accurate results can be obtained on a personal computer for relatively large subsystem sizes.

5 Apr 2022

A general framework for randomized benchmarking

J. Helsen, I. Roth, E. Onorati, A. H. Werner and J. Eisert

PRX Quantum 3, 020357 (2022)

Randomized benchmarking (RB) refers to a collection of protocols that in the past decade have become central methods for characterizing quantum gates. These protocols aim at efficiently estimating the quality of a set of quantum gates in a way that is resistant to state preparation and measurement errors. Over the years many versions have been developed, however, a comprehensive theoretical treatment of RB has been missing. In this work, we develop a rigorous framework of RB general enough to encompass virtually all known protocols as well as novel, more flexible extensions. Overcoming previous limitations on error models and gate sets, this framework allows us, for the first time, to formulate realistic conditions under which we can rigorously guarantee that the output of any RB experiment is well-described by a linear combination of matrix exponential decays. We complement this with a detailed analysis of the fitting problem associated with RB data. We introduce modern signal processing techniques to RB, prove analytical sample complexity bounds, and numerically evaluate performance and limitations. In order to reduce the resource demands of this fitting problem, we introduce novel, scalable post-processing techniques to isolate exponential decays, significantly improving the practical feasibility of a large set of RB protocols. These post-processing techniques overcome shortcomings in efficiency of several previously proposed methods such as character benchmarking and linear-cross entropy benchmarking. Finally, we discuss, in full generality, how and when RB decay rates can be used to infer quality measures like the average fidelity. On the technical side, our work substantially extends the recently developed Fourier-theoretic perspective on RB by making use of the perturbation theory of invariant subspaces, as well as ideas from signal processing.


Reducing 2-qubit gate count for ZX-calculus based quantum circuit optimization

K. Staudacher, T. Guggemos, W. Gehrke, and S. Grundner-Culemann

Quantum Processing and Languages (QPL22), 27. Jun. - 01. Jul. 2022, Oxford, UK

In the near term, programming quantum computers will remain severely limited by low quantum volumes. Therefore, it is desirable to implement quantum circuits with the fewest resources possible. For the common Clifford+T circuits, most research is focused on reducing the number of T gates, since they are an order of magnitude more expensive than Clifford gates in quantum error corrected encoding schemes. However, this optimization sometimes leads to more 2-qubit gates, which, even though they are less expensive in terms of fault-tolerance, contribute significantly to the overall circuit cost. Approaches based on the ZX-calculus have recently gained some popularity in the field, but reduction of 2-qubit gates is not their focus. In this work, we present an alternative for improving 2-qubit gate count of a quantum circuit with the ZX-calculus by using heuristics in ZX-diagram simplification. Our approach maintains the good reduction of the T gate count provided by other strategies based on ZX-calculus, thus serving as an extension for other optimization algorithms. Our results show that combining the available ZX-calculus-based optimizations with our algorithms can reduce the number of 2-qubit gates by as much as 40 % compared to current approaches using ZX-calculus. Additionally, we improve the results of the best currently available optimization technique of Nam et. al [22] for some circuits by up to 15 %


Bringing quantum acceleration to supercomputers

M. Ruefenacht, B. G. Taketani, P. Lähteenmäki, V. Bergholm, D. Kranzlmüller, L. Schulz, M. Schulz

White paper

In a world shifting towards sustainable growth, high-performance computing has an important challenge: delivering on a growing demand for increased computational power, while keeping energy consumption at bay. Quantum computers promise to meet these challenges with an exponential performance improvement for key applications and are anticipated to be the next big technological breakthrough in the field. This paper discusses part of the road ahead to integrate quantum acceleration into supercomputers, as well as the critical steps and decisions required in order to build the quantum future of high-performance computing and make important strides towards green computing.