Our group studies quantum many-body physics with semidefinite programming. Many questions about large quantum systems, such as whether their correlations are nonlocal, how entangled they are or how large a spectral gap is, can be relaxed to convex optimisation problems that a computer solves efficiently, and symmetries keep these problems tractable as the system grows. We use this to look for Bell correlations in many-body systems, to certify many-body properties from partial information and to bound spectral gaps.

We also work on quantum algorithms for near-term and distributed quantum computers, device-independent certification, quantum machine learning, metrology, entanglement theory and thermodynamics.

Our group is embedded in Applied Quantum Algorithms (aQa), a team in Leiden with members at the institutes of physics, computer science, chemistry and mathematics. If you are considering a PhD or a postdoc with us, the Working together section of the contact page describes the current opportunities and how to apply.

Ground-state preparation. Our general-purpose quantum algorithm prepares the ground state of a Hamiltonian from a trial state, using techniques developed for the quantum linear systems problem. Compared with phase estimation, its runtime is exponentially better in the allowed error, and it needs fewer ancilla qubits¹.

Adiabatic algorithms. We proposed a way to probe the spectrum along an adiabatic evolution, and a variational quantum adiabatic algorithm that optimises the path². Experiments with Rydberg atom arrays have since used such an algorithm³. Our adiabatic echo verification protocol evolves forward and then backward, and mitigates errors from both non-adiabatic transitions and hardware noise⁴.

Eigenstate preparation on connected devices. We showed that two devices sharing entanglement through a coherent link can prepare the same eigenstate with shallower circuits⁵. Our later distributed algorithm, inspired by iterative phase estimation, reduces the energy variance faster than single-device implementations⁶. Part of the motivation came from the SuperQuLAN project (see also distributed quantum computing below).

Differential equations. We derived analytical error and resource estimates for variational quantum algorithms that integrate a differential equation step by step with a classical Runge–Kutta method. Our option pricing example, whose equation is derived from a stochastic differential equation, shows that the method applies beyond ordinary differential equations (ODEs)⁷. For continuous-variable quantum computers, we gave an algorithm that solves arbitrary one-dimensional polynomial ODEs⁸.

Gradients and tomography. For a generator with a known spectrum, however close its eigenvalues are, we proposed parameter shift rules for any linear combination of k-th derivatives, and discussed how to choose the shifts to minimise their number and the variance of the estimator⁹. We reconstructed reduced density matrices with a semidefinite program that imposes relaxed compatibility constraints between overlapping ones, which, for the same number of measurements, gives on average tighter energy bounds than unconstrained tomography. In simulations it also improved an algorithmic cooling procedure¹⁰. An earlier workshop paper outlines a roadmap towards algorithmic cooling on NISQ devices¹¹.

Shallow circuits and circuit cutting. Our reduce&chop method lets repeated runs of a shallower device mimic a deeper computation in certain cases of interest, by chopping the circuit in two and adding a shallow variational circuit¹². We also introduced multiple-basis representations, linear combinations of states that are each sparse in a different basis, which include matrix-product states and stabilizer states¹³. And we showed that no locally acting circuit cutting scheme (all existing ones are) can efficiently separate even a single qubit from the rest of a circuit unless BPP = BQP¹⁴.

High-energy physics and quantum software. We contributed to the QC4HEP roadmap on quantum computing for high-energy physics, led by CERN, DESY and IBM, with benchmark applications that could be addressed in the near future¹⁵. The QANSWER 2020 workshop on quantum software engineering, whose proceedings include the workshop paper above, led to the Talavera Manifesto¹⁶.

Review of the field. We reviewed recent progress in distributed quantum information processing, covering protocols and the platforms and algorithms that realise them¹⁷. If you are considering a project with us, start there.

Bell sampling on networks. Noisy graph states can be characterised by their diagonal entries in the graph-state basis. We proved that Bell-basis measurements on two copies estimate all of them from O(n) samples, against O(2ⁿ) for the best known single-qubit method¹⁸.

Entanglement from symmetry tests. Party-wise swap tests on two copies of a pure state give its concentratable entanglement¹⁹. We generalised it to a class of multipartite entanglement monotones, estimated from permutation tests²⁰. We also showed that symmetry tests give a family of well-defined entanglement measures²¹.

Projecting before measuring. Joint measurements on several copies give access to nonlinear quantities such as the purity, but standard protocols become unfeasible for large systems. We proposed to project the copies at random onto a smaller subspace, which saves coherent quantum resources at the cost of a sampling overhead²².

Learning to distribute entanglement. In quantum repeaters, link generation and entanglement swapping are often probabilistic. We used reinforcement learning, with an interpretable model, to optimise centralised policies under classical communication delays, and introduced a fixed local policy²³. At high success probabilities, both are faster than the straightforward generalisation of swap-as-soon-as-possible to this setting.

Devices as black boxes. Device-independent protocols draw conclusions about quantum devices from their input and outcome statistics alone. Violating a Bell inequality certifies, within quantum theory, that the devices share an entangled state. We design Bell inequalities to certify states, entanglement depth and randomness.

Self-testing with Bell inequalities. In self-testing, observing the maximal quantum violation identifies the only state and measurements that reach it, up to local isometries. The SATWAP inequalities we introduced, with any number of measurements and outcomes, are maximally violated by maximally entangled states²⁴, and their multipartite version by GHZ states of any number of parties and local dimension²⁵. Our generalisation of the CHSH inequality to an odd prime number of measurements and outcomes is maximally violated by the maximally entangled state and mutually unbiased bases, and self-tests both for qutrits²⁶.

Robust self-tests and many-body states. Real data need robust self-tests, where a near-maximal violation still certifies a nearly ideal state. For graph states we built such Bell inequalities, whose number of correlators grows only linearly with the number of parties²⁷. In a steering scenario we gave robust self-tests of the remotely prepared states from several standard Bell inequalities²⁸. Our Bell inequalities for the qudit toric code self-test the whole qutrit code subspace, up to complex conjugation, to our knowledge the first self-test of a qutrit subspace²⁹.

Entanglement and Bell correlation depth. Entanglement depth counts the particles in the largest genuinely entangled group. The two-body Bell inequalities described below also witness it device-independently, from second moments of collective measurements alone³⁰, with details in a companion paper³¹. Our witnesses of Bell correlation depth, the analogue for nonlocality, can reveal genuine nonlocality in any number of particles from two trusted collective measurements, where multipartite expectation values can be evaluated³². Existing data from a Bose–Einstein condensate of 480 atoms³³ already violate our two-body witness for depth 6.

Randomness certification. We showed that in any no-signalling theory the violation of certain Bell inequalities tightly bounds what an outsider can learn about a measurement outcome, and linked these monogamy relations to randomness generation and amplification³⁴. We proved, and demonstrated with our collaborators in a photonic experiment, that violating any two-input Bell inequality is necessary and sufficient to certify randomness, while with more inputs some nonlocal states cannot³⁵. For two parties we gave a necessary and sufficient condition for certifying randomness, in terms of measurement incompatibility³⁶.

Experimental tests. With experimental colleagues we used the SATWAP inequalities on a silicon-photonics chip that generates entangled states of dimension up to 15 × 15, for self-testing entangled two-qutrit states and device-independent randomness expansion³⁷. More recently, with an experimental team, we certified genuine multipartite Bell-operator correlations for up to 24 qubits on a programmable superconducting processor³⁸.

Nonlocality in large systems. Correlations that violate a Bell inequality admit no local hidden variable model and are the resource behind device-independent protocols. Standard multipartite inequalities need many-particle correlations, but many-body experiments typically measure one- and two-body correlators or the energy. This line of work began during my PhD³⁹ and asks whether such quantities are enough to reveal Bell correlations. Our Varenna lecture notes review entanglement and nonlocality in many-body systems⁴⁰.

Two-body Bell inequalities. We constructed permutationally invariant Bell inequalities from one- and two-body correlators, violated for any number of parties and testable by measuring total spin components⁴¹. A longer paper studies their tightness and maximal violation and adds classes violated by Dicke states⁴². Witnesses derived from these inequalities soon revealed Bell correlations in spin-squeezed states of a Bose-Einstein condensate of about 480 atoms⁴³ and of an ensemble of 5×10⁵ atoms⁴⁴. We also showed that any translationally invariant Bell inequality can be maximally violated by a translationally invariant state⁴⁵.

The energy as a Bell witness. To one-dimensional interacting Hamiltonians we assigned Bell inequalities whose quantum value, for suitable observables, corresponds to the energy, so that low enough energy certifies nonlocality⁴⁶. We later computed the classical bounds of broad families of Bell inequalities by contracting tensor networks in tropical algebra, also in the thermodynamic limit for some translationally invariant systems⁴⁷. In one dimension and with finite-range correlators, the translationally invariant local polytope has a number of vertices bounded independently of the system size⁴⁸.

Lattices and quantum processors. On square lattices, we related the topology set by the boundary conditions to how well the assigned Bell inequality reveals nonlocality⁴⁹. Since the mapping between inequalities and Hamiltonians works both ways, we built an optimisation loop on it⁵⁰ and, together with experimental colleagues, applied it to a honeycomb lattice of more than 70 superconducting qubits.

Bounding the local set. For many parties, the polytope of local correlations is too complex to describe in full. We relaxed its defining conditions into a convergent hierarchy of semidefinite programs whose size does not depend on the number of parties⁵¹. For spin-1 atoms and other three-level particles, we gave strategies to derive three-outcome Bell inequalities⁵², and a dimension witness that certifies genuine three-level correlations in systems of any size⁵³.

Temperature and quantum chaos. We showed that spin systems with infinite-range interactions can violate a Bell inequality at thermal equilibrium, up to a finite critical temperature⁵⁴. We introduced a Bell inequality for many three-level systems and studied its Bell operator as a Hamiltonian⁵⁵. In every symmetry sector that shows nonlocality, the maximally violating settings give the Poissonian level statistics of integrable systems, and generic settings the Wigner-Dyson statistics of chaotic ones. We dedicated this paper to the memory of Fritz Haake.

More correlators and settings. With up to four-body correlators, we presented about twenty families of permutationally invariant inequalities that tolerate more noise than known ones or detect Bell correlations in highly non-Gaussian spin states⁵⁶. We observed numerically that local maxima of the ratio between quantum value and classical bound typically correspond to facets of the local polytope, which allows an iterative gradient-based search for tight inequalities⁵⁷. For two-body families that we built with any number of measurement settings, more settings give stronger violations⁵⁸. Together with experimental colleagues, we certified Bell correlations in postselected six- and eight-photon states close to Dicke states⁵⁹.

Stoquastic Bell operators. Read as a Hamiltonian, a Bell operator is stoquastic in a given basis when its off-diagonal elements are nonpositive. We introduced the stoquasticity cone to characterise when permutationally invariant Bell operators are stoquastic, and showed that with two two-outcome measurements per party and at most three-body correlators they can always be made so⁶⁰. In the thermodynamic limit, the cone has an exact semidefinite description⁶¹.

Preparing and verifying states. A tensor network state is a ground state of a parent Hamiltonian, which suggests preparing it on a quantum computer by adiabatic evolution, at a speed set by the spectral gap. For a family of such states on regular lattices that can be prepared efficiently this way, we showed that semidefinite programming gives efficiently computable lower bounds on the gap. We also derived observables that characterise the state uniquely, for verification⁶².

Optimising the gap. The parent Hamiltonian is not unique, and we used this freedom to maximise the spectral gap, which gives an efficient adiabatic algorithm to prepare tensor network states⁶³. It applies to injective and non-injective tensors, and we demonstrated it on random one-dimensional states, the AKLT ground state and the GHZ state.

Certifying spectral gaps. We formulated gap certification for frustration-free Hamiltonians in the thermodynamic limit as a hierarchy of semidefinite programs, whose lower bound improves with the level⁶⁴. Knabe's bound⁶⁵ and its later improvements⁶⁶ are particular solutions, so the hierarchy matches or surpasses them. On one-dimensional spin chains we observed improvements of several orders of magnitude.

Expressivity of quantum circuits. We showed that parametrised quantum circuits (PQCs) can approximate continuous functions, p-integrable functions and the Sobolev spaces H^k under specific distances⁶⁷. We also proved that several PQC architectures are universal for generating continuous multivariate distributions, and our resource bounds show a trade-off between qubits and measurements⁶⁸.

Learning from noisy quantum data. We trained models of a density functional for the Fermi-Hubbard model on the kind of noisy data that quantum algorithms produce⁶⁹. A neural network can generalise from small noisy datasets and filter out unbiased sampling noise, so the model outperforms any individual training point.

Learning continuous-variable states. For continuous-variable systems, we introduced kernel quantum state estimation, a non-parametric method that reconstructs a state and quantities such as its purity from noisy data, without prior knowledge of the state⁷⁰. It also handles multimodal, non-Gaussian states.

Generative models for finance. We investigated whether quantum correlations help quantum-inspired generative adversarial networks to generate financial time series⁷¹. The trained models produce series that match the target distribution and the desired temporal correlations, with a quality that depends on the hyperparameters and the simulation method.

Symmetric many-body states. The quantum marginal problem asks whether few-body reduced states can come from one global state. For symmetric states we gave a method to solve it with an efficient semidefinite program⁷². We also lower-bounded the fidelity with a target state from few-body observables, so spin-squeezed states can be certified from collective measurements⁷³.

Relaxations and reinforcement learning. We combined relaxations with deep reinforcement learning to find the best bounds within a limited computational budget⁷⁴. As an example, we bounded many-body ground-state energies from below.

Bounds for lattice gauge theories. Variational methods only give upper bounds on the ground-state energy of lattice gauge theories. We built a hierarchy of semidefinite programs that gives increasingly good lower bounds and propagates them to observables such as Wilson loops⁷⁵. The method can also certify results of quantum simulators.

Few-body ground states as probes. In quantum metrology, states at the Heisenberg limit typically need full-body correlators to be prepared, but experimental Hamiltonians are often few-body. We found that typical random symmetric ground states of k-body permutation-invariant Hamiltonians show Heisenberg scaling. We also established a trade-off between the Hamiltonian's gap, which quantifies how hard its ground state is to prepare, and the state's quantum Fisher information⁷⁶. Our work on many-body Bell correlations faces the same restriction to few-body correlators.

Topology of graph states. The quantum Fisher information of a graph state counts pairs of vertices with identical neighbourhoods. We asked whether the Betti numbers of the graph's clique complex tell when the state beats the standard quantum limit. Maximising a Betti number drives random graphs towards multipartite structures whose quantum Fisher information lies well beyond that limit, but Betti numbers alone do not determine it⁷⁷.

Symmetric states. Much of our work here concerns states that are unchanged when particles are permuted. Our review⁷⁸ covers how tomography and Bell tests adapt to them. Its part on structure and properties is pedagogical, so students can start there.

PPT entangled symmetric states. Entangled states that pass the PPT criterion are bound entangled. We showed that four-qubit symmetric states can be PPT across every bipartition and still be entangled⁷⁹, and gave separability criteria for any number of qubits⁸⁰.

Copositive witnesses. For two parties, we recast the separability of mixtures of Dicke states as a conic optimisation problem, with copositive matrices as witnesses⁸¹. PPT then decides it up to local dimension four, but not beyond. Extended copositive matrices later gave witnesses beyond the diagonal case, and new bound entangled symmetric states⁸².

Optimal witnesses. In my first paper we proved two conditions equivalent to optimality for decomposable entanglement witnesses on a qubit and an n-level system⁸³. We later showed that a proposed counterexample to the structural physical approximation conjecture is not optimal, so it does not disprove the conjecture⁸⁴.

Entanglement versus nonlocality. For any number of parties, we constructed genuinely multipartite entangled states that are not genuinely multipartite nonlocal⁸⁵. Our later generalisation also gives genuinely entangled states that are unsteerable across some bipartitions⁸⁶.

Positive maps and entanglement depth. In a paper in memory of Andrzej Kossakowski, we used positive maps that are not completely positive to certify upper bounds on the entanglement depth of N-qubit states⁸⁷. We also suggested a similar test of compatibility with local hidden-variable models.

Unextendible product operator bases. We generalised unextendible product bases, which cannot be perfectly distinguished by LOCC, from states to operators⁸⁸. Some of the resulting bases share this property, and we showed how to distinguish one with three ebits.

Sector lengths of graph states. The sector length distribution of a graph state is hard to compute in general. We encode the distributions of whole graph families, such as paths, cycles, stars and grids, in a generating function⁸⁹, which gives them in closed form.

Detecting magic. Stabilizer circuits can be efficiently simulated classically, so states that are not mixtures of stabilizer states (they have magic) are a resource. We show that detecting magic from a limited set of Pauli measurements is efficient when their frustration graph is perfect and no sign dependencies are active⁹⁰.

Most energetic passive states. No cyclic unitary process can extract work from a passive state. Thermal states are the passive states with the least energy at fixed entropy. We found those with the most, showed that they also minimise the entropy at fixed energy, and derived thermodynamic bounds for finite-dimensional quantum systems from them⁹¹.

Fast thermal state preparation. Thermal states can be prepared by letting a system relax under dissipative dynamics. We gave a systematic framework to construct an optimal Lindbladian for a given Hamiltonian and demonstrated it on the kinetic Ising model and the Lipkin–Meshkov–Glick model⁹².

The BIG Bell Test. A Bell test needs measurement settings that nobody can predict, and Bell argued that human free choices could supply them. I was part of the BIG Bell Test collaboration, which put this idea into practice⁹³. About 100,000 people played an online video game, and their choices fixed the measurement settings of 13 experiments on five continents. The observed correlations contradict local realism.

Writing for mathematicians. For the December 2022 newsletter of IMTech, the Institute of Mathematics of UPC-BarcelonaTech, I wrote a short note on Bell correlations in quantum many-body systems. It shows how permutation symmetry and a semidefinite relaxation give a test whose cost does not grow with the number of parties. The same methods are behind our work on many-body Bell correlations.

Visualising multi-qubit states. We proposed a mapping that represents a set of arbitrary multi-qubit pure states uniquely on a binary tree of Bloch spheres, by combining the Schmidt decomposition with the Bloch-sphere representation⁹⁴. For a time evolution, the tree shows its periodicity and some of its entanglement properties at a glance. The code and a Python visualisation library are open source.

Commentaries on classical simulation. I have written three commentaries on new classical simulation methods. In Physics⁹⁵ I discussed work showing that an imperfect quantum computer can be simulated classically at a tiny fraction of the cost of a perfect one, by compressing its state into a matrix product state⁹⁶. The second, in Nature Computational Science⁹⁷, was on splitting a tensor network contraction into smaller parallel subtasks, which speeds up the classical simulation of quantum circuits⁹⁸. The third, in Nature Physics⁹⁹, was on a tensor network algorithm for Gaussian boson sampling whose cost drops when photon loss is high, and which challenges recent optical experiments that claimed quantum advantage¹⁰⁰.

Teleportation in the classroom. With Barry Fitzgerald and Patrick Emonts we wrote an article for Physics Education on teaching quantum teleportation with the help of Santa Claus¹⁰¹. It is meant for teachers in secondary school and early university, and outlines a classroom worksheet built around common naive conceptions about quantum physics.