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Jonas Helsen

Publications and source records attributed to Jonas Helsen.

At least 19 recordsLinked to original sources
arXiv2026

(Almost) quadruply optimal unitary designs in 1D

We construct $n$-qubit approximate unitary $k$-designs in 1D systems, achieving circuit depth $O(\log(n/\varepsilon) + k\log k)$ with relative error $\varepsilon$ and requiring $O(nk\log k)$ magic gates. This matches existing lower bounds $\Omega(\log(n/\varepsilon) + k)$ for circuit depth, and $\widetilde{\Omega}(nk)$ for the required number of $T$ gates, up to a $\log k$ factor, achieving simultaneous near-optimality in all parameters. Our construction is based on a combination and refinement of two existing results. We reduce the required magic block size for breaking Clifford symmetries in the magic-augmented circuit construction of Zhang et al. from $O(k\log k)$ to $O(\log k)$. We also improve the breakthrough construction of Chen et al. to construct a generating set of 1D local constant-depth circuits for the unitary group with a constant spectral gap, making $O(\log k)$-local random unitaries realizable in depth $O(k\log k)$. As a by-product, we provide a constant-size 1D-local generating set for the Clifford group, which we expect to be of independent interest. Combining the two results with the gluing lemma, we prove the final result.

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arXiv2026

On Clifford hierarchy testing and near-extremizers of noncommutative uniformity norms

We consider the problem of testing whether an unknown unitary is close to a specified level of the Clifford hierarchy. Bu, Gu, and Jaffe proposed a candidate tester for this task based on a connection with noncommutative analogues of the Gowers uniformity norms. The complexity of this tester -- whose analysis depends on a robust characterization of the near-extremizers of these norms -- was left open. We establish such a characterization for the fourth noncommutative uniformity norm and, as a consequence, obtain an efficient tester for the third level of the Clifford hierarchy. We further discuss possible routes toward resolving the problem of testing for all higher levels, highlighting the main barriers that remain.

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arXiv2026

Arts & crafts: Strong random unitaries and geometric locality

We study the problem of constructing strong approximate unitary $k$-designs on $D$-dimensional grids (and more generally on Cartesian products of graphs), building on the work of Schuster et al. arXiv:2509.26310 which establishes strong unitary designs in 1D and in all-to-all connectivity. We provide two constructions. The first construction leverages the existing all-to-all connectivity result with general routing theory to provide flexible (but slightly suboptimal) strong $k$-designs in arbitrary connectivities. The second construction is more direct, requires no auxiliaries and has provably optimal depth (in the number of qubits $n$) for $D$-dimensional grids with constant dimension. Combining these techniques also allows us to construct strong pseudorandom unitaries on $D$-dimensional grids with provably optimal depth.

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arXiv2026

Error correction with brickwork Clifford circuits

We prove that random 1D Clifford brickwork circuits form (in expectation) good approximate quantum error correction codes in logarithmic depth. Our proof makes use of the statistical mechanics techniques for random circuits developed by Dalzell et al. [PRX Quantum 3, 010333], adapted extensively to our own purpose. We also consider exact error correction, where we give matching upper and lower bounds for the required depth in which random 1D Clifford brickwork circuits become error correcting.

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arXiv2026

A Bravyi-K\"onig theorem for Floquet codes generated by locally conjugate instantaneous stabiliser groups

The Bravyi-K\"onig (BK) theorem is an important no-go theorem for the dynamics of topological stabiliser quantum error correcting codes. It states that any logical operation on a $D$-dimensional topological stabiliser code that can be implemented by a short-depth circuit acts on the codespace as an element of the $D$-th level of the Clifford hierarchy. In recent years, a new type of quantum error correcting codes based on Pauli stabilisers, dubbed Floquet codes, has been introduced. In Floquet codes, syndrome measurements are arranged such that they dynamically generate a codespace at each time step. Here, we show that the BK theorem holds for a definition of Floquet codes based on locally conjugate stabiliser groups. Moreover, we introduce and define a class of generalised unitaries in Floquet codes that need not preserve the codespace at each time step, but that combined with the measurements constitute a valid logical operation. We derive a canonical form of these generalised unitaries and show that the BK theorem holds for them too.

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arXiv2025

Anti-concentration is (almost) all you need

Until very recently, it was generally believed that the (approximate) 2-design property is strictly stronger than anticoncentration of random quantum circuits, mainly because it was shown that the latter anticoncentrate in logarithmic depth, while the former generally need linear depth circuits. This belief was disproven by recent results which show that so-called relative-error approximate unitary designs can, in fact, be generated in logarithmic depth, implying anticoncentration. Their result does however not apply to ordinary local random circuits, a gap which we close in this letter, at least for 2-designs. More precisely, we show that anticoncentration of local random quantum circuits already implies that they form relative-error approximate state 2-designs, making them equivalent properties for these ensembles. Our result holds more generally for any random circuit which is invariant under local (single-qubit) unitaries, independent of the architecture.

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arXiv2025

Clifford testing: algorithms and lower bounds

We consider the problem of Clifford testing, which asks whether a black-box $n$-qubit unitary is a Clifford unitary or at least $\varepsilon$-far from every Clifford unitary. We give the first 4-query Clifford tester, which decides this problem with probability $\mathrm{poly}(\varepsilon)$. This contrasts with the minimum of 6 copies required for the closely-related task of stabilizer testing. We show that our tester is tolerant, by adapting techniques from tolerant stabilizer testing to our setting. In doing so, we settle in the positive a conjecture of Bu, Gu and Jaffe, by proving a polynomial inverse theorem for a non-commutative Gowers 3-uniformity norm. We also consider the restricted setting of single-copy access, where we give an $O(n)$-query Clifford tester that requires no auxiliary memory qubits or adaptivity. We complement this with a lower bound, proving that any such, potentially adaptive, single-copy algorithm needs at least $\Omega(n^{1/4})$ queries. To obtain our results, we leverage the structure of the commutant of the Clifford group, obtaining several technical statements that may be of independent interest.

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arXiv2025

Will it glue? On short-depth designs beyond the unitary group

We study the formation of short-depth designs beyond the unitary group. We provide a range of results on several groups of broad interest in quantum information science: the Clifford group, the orthogonal group, the unitary symplectic groups, and the matchgate group. For all of these groups, we prove that analogues of unitary designs cannot be generated by any circuit ensemble with light-cones that are smaller than the system size. This implies linear lower bounds on the circuit depth in one-dimensional systems. For the Clifford and orthogonal group, we moreover show that a broad class of circuits cannot generate designs in sub-linear depth on any circuit architecture. We show this by exploiting observables in the higher-order commutants of each group, which allow one to distinguish any short-depth circuit from truly random. While these no-go results rule out short-depth unitary designs, we prove that slightly weaker forms of randomness -- including additive-error state designs and anti-concentration in sampling distributions -- nevertheless emerge at logarithmic depths in many cases. Our results reveal that the onset of randomness in shallow quantum circuits is a widespread yet subtle phenomenon, dependent on the interplay between the group itself and the context of its application.

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arXiv2025

Minimising the number of edges in LC-equivalent graph states

Graph states are a powerful class of entangled states with numerous applications in quantum communication and quantum computation. Local Clifford (LC) operations that map one graph state to another can alter the structure of the corresponding graphs, including changing the number of edges. Here, we tackle the associated edge-minimisation problem: finding graphs with the minimum number of edges in the LC-equivalence class of a given graph. Such graphs are called minimum edge representatives (MER) and are crucial for minimising the resources required to create a graph state. We leverage Bouchet's algebraic formulation of LC-equivalence to encode the edge-minimisation problem as an integer linear program (EDM-ILP). We further propose a simulated annealing (EDM-SA) approach guided by the local clustering coefficient for edge minimisation. We identify new MERs for graph states with up to 16 qubits by combining EDM-SA and EDM-ILP. We extend the ILP to weighted-edge minimisation, where each edge has an associated weight, and prove that this problem is NP-complete. Finally, we employ our tools to minimise the resources required to create all-photonic generalised repeater graph states using fusion operations.

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arXiv2025

Catalytic entanglement transformations with noisy hardware

The availability of certain entangled resource states (catalyst states) can enhance the rate of converting several less entangled states into fewer highly entangled states in a process known as catalytic entanglement concentration (EC). Here, we extend catalytic EC from pure states to mixed states and numerically benchmark it against non-catalytic EC and distillation in the presence of state-preparation errors and operational errors. Furthermore, we analyse the re-usability of catalysts in the presence of such errors. To do this, we introduce a novel recipe for determining the positive-operator valued measurements (POVM) required for EC transformations, which allows for making tradeoffs between the number of communication rounds and the number of auxiliary qubits required. We find that in the presence of low operational errors and depolarising noise, catalytic EC can provide better rates than distillation and non-catalytic EC.

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arXiv2025

Trotter error and gate complexity of the SYK and sparse SYK models

The Sachdev--Ye--Kitaev (SYK) model is a prominent model of strongly interacting fermions that serves as a toy model of quantum gravity and black hole physics. In this work, we study the Trotter error and gate complexity of the quantum simulation of the SYK model using Lie--Trotter--Suzuki formulas. Building on recent results by Chen and Brand\~{a}o (arXiv:2111.05324), we derive bounds on the first- and higher-order Trotter error of the SYK model, and subsequently find near-optimal gate complexities for simulating these models. For the $k$-local SYK model on $n$ Majorana fermions, at time $t$, our gate complexity estimates for the first-order Lie--Trotter--Suzuki formula scales with $\tilde{\mathcal{O}}(n^{k+\frac{5}{2}}t^2)$ for even $k$ and $\tilde{\mathcal{O}}(n^{k+3}t^2)$ for odd $k$, and the gate complexity of simulations using higher-order formulas scales with $\tilde{\mathcal{O}}(n^{k+\frac{1}{2}}t)$ for even $k$ and $\tilde{\mathcal{O}}(n^{k+1}t)$ for odd $k$. Given that the SYK model has $\Theta(n^k)$ terms, these estimates are close to optimal. These gate complexities can be further improved upon in the context of simulating the time evolution of an arbitrary fixed input state $|\psi\rangle$, leading to a $\mathcal{O}(n^2)$-reduction in gate complexity for first-order and $\mathcal{O}(\sqrt{n})$-reduction for higher-order formulas. Regarding the sparse SYK model, which is a simplified variant of the SYK model obtained by deleting all but a $\Theta(n)$ fraction of the terms in a uniformly i.i.d. manner, our average gate complexity estimates for higher-order formulas scale as $\tilde{\mathcal{O}}(n^{1+\frac{1}{2}} t)$ for even $k$ and $\tilde{\mathcal{O}}(n^{2} t)$ for odd $k$. Similar to the full SYK model, we obtain a $\mathcal{O}(\sqrt{n})$-reduction simulating the time evolution of an arbitrary fixed input state $|\psi\rangle$.

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arXiv2024

Random regular graph states are complex at almost any depth

Graph states are fundamental objects in the theory of quantum information due to their simple classical description and rich entanglement structure. They are also intimately related to IQP circuits, which have applications in quantum pseudorandomness and quantum advantage. For us, they are a toy model to understand the relation between circuit connectivity, entanglement structure and computational complexity. In the worst case, a strict dichotomy in the computational universality of such graph states appears as a function of the degree $d$ of a regular graph state [GDH+23]. In this paper, we study the average-case complexity of simulating random graph states of varying degree when measured in random product bases and give distinct evidence that a similar complexity-theoretic dichotomy exists in the average case. Specifically, we consider random $d$-regular graph states and prove three distinct results: First, we exhibit two families of IQP circuits of depth $d$ and show that they anticoncentrate for any $2 < d = o(n^{1/2})$ when measured in a random $X$-$Y$-plane product basis. This implies anticoncentration for random constant-regular graph states. Second, in the regime $d = \Theta(n^c)$ with $c \in (0,1)$, we prove that random $d$-regular graph states contain polynomially large grid graphs as induced subgraphs with high probability. This implies that they are universal resource states for measurement-based computation. Third, in the regime of high degree ($d\sim n/2$), we show that random graph states are not sufficiently entangled to be trivially classically simulable, unlike Haar random states. Proving the three results requires different techniques -- the analysis of a classical statistical-mechanics model using Krawtchouck polynomials, graph theoretic analysis using the switching method, and analysis of the ranks of submatrices of random adjacency matrices, respectively.

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arXiv2024

Single-copy stabilizer testing

We consider the problem of testing whether an unknown $n$-qubit quantum state $|\psi\rangle$ is a stabilizer state, with only single-copy access. We give an algorithm solving this problem using $O(n)$ copies, and conversely prove that $\Omega(\sqrt{n})$ copies are required for any algorithm. The main observation behind our algorithm is that when repeatedly measuring in a randomly chosen stabilizer basis, stabilizer states are the most likely among the set of all pure states to exhibit linear dependencies in measurement outcomes. Our algorithm is designed to probe deviations from this extremal behavior. For the lower bound, we first reduce stabilizer testing to the task of distinguishing random stabilizer states from the maximally mixed state. We then argue that, without loss of generality, it is sufficient to consider measurement strategies that a) lie in the commutant of the tensor action of the Clifford group and b) satisfy a Positive Partial Transpose (PPT) condition. By leveraging these constraints, together with novel results on the partial transposes of the generators of the Clifford commutant, we derive the lower bound on the sample complexity.

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arXiv2024

Mutual information fluctuations and non-stabilizerness in random circuits

The emergence of quantum technologies has brought much attention to the characterization of quantum resources as well as the classical simulatability of quantum processes. Quantum resources, as quantified by non-stabilizerness, have in one theoretical approach been linked to a family of entropic, monotonic functions. In this work, we demonstrate both analytically and numerically a simple relationship between non-stabilizerness and information scrambling using the fluctuations of an entropy-based quantifier. Specifically, we find that the non-stabilizerness generated by a random quantum circuit is proportional to fluctuations of mutual information. Furthermore, we explore the role of non-stabilizerness in measurement-induced entanglement phase transitions. We find that the fluctuations of mutual information decrease with increasing non-stabilizerness yielding potentially easier identification of the transition point. Our work establishes a key connection between quantum resource theory, information scrambling and measurement-induced entanglement phase transitions.

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arXiv2024

Quantum PCPs: on Adaptivity, Multiple Provers and Reductions to Local Hamiltonians

We define a general formulation of quantum PCPs, which captures adaptivity and multiple unentangled provers, and give a detailed construction of the quantum reduction to a local Hamiltonian with a constant promise gap. The reduction turns out to be a versatile subroutine to prove properties of quantum PCPs, allowing us to show: (i) Non-adaptive quantum PCPs can simulate adaptive quantum PCPs when the number of proof queries is constant. In fact, this can even be shown to hold when the non-adaptive quantum PCP picks the proof indices simply uniformly at random from a subset of all possible index combinations, answering an open question by Aharonov, Arad, Landau and Vazirani (STOC '09). (ii) If the $q$-local Hamiltonian problem with constant promise gap can be solved in $\mathsf{QCMA}$, then $\mathsf{QPCP}[q] \subseteq \mathsf{QCMA}$ for any $q \in O(1)$. (iii) If $\mathsf{QMA}(k)$ has a quantum PCP for any $k \leq \text{poly}(n)$, then $\mathsf{QMA}(2) = \mathsf{QMA}$, connecting two of the longest-standing open problems in quantum complexity theory. Moreover, we also show that there exist (quantum) oracles relative to which certain quantum PCP statements are false. Hence, any attempt to prove the quantum PCP conjecture requires, just as was the case for the classical PCP theorem, (quantumly) non-relativizing techniques.

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arXiv2023

Optimizing sparse fermionic Hamiltonians

We consider the problem of approximating the ground state energy of a fermionic Hamiltonian using a Gaussian state. In sharp contrast to the dense case, we prove that strictly $q$-local $\rm {\textit {sparse}}$ fermionic Hamiltonians have a constant Gaussian approximation ratio; the result holds for any connectivity and interaction strengths. Sparsity means that each fermion participates in a bounded number of interactions, and strictly $q$-local means that each term involves exactly $q$ fermionic (Majorana) operators. We extend our proof to give a constant Gaussian approximation ratio for sparse fermionic Hamiltonians with both quartic and quadratic terms. With additional work, we also prove a constant Gaussian approximation ratio for the so-called sparse SYK model with strictly $4$-local interactions (sparse SYK-$4$ model). In each setting we show that the Gaussian state can be efficiently determined. Finally, we prove that the $O(n^{-1/2})$ Gaussian approximation ratio for the normal (dense) SYK-$4$ model extends to SYK-$q$ for even $q>4$, with an approximation ratio of $O(n^{1/2 - q/4})$. Our results identify non-sparseness as the prime reason that the SYK-$4$ model can fail to have a constant approximation ratio.

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arXiv2023

Fermionic Hamiltonians without trivial low-energy states

We construct local fermionic Hamiltonians with no low-energy trivial states (NLTS), providing a fermionic counterpart to the NLTS theorem. Distinctly from the qubit case, we define trivial states via finite-depth $\textit{fermionic}$ quantum circuits. We furthermore allow free access to Gaussian fermionic operations, provided they involve at most $O(n)$ ancillary fermions. The desired fermionic Hamiltonian can be constructed using any qubit Hamiltonian which itself has the NLTS property via well-spread distributions over bitstrings, such as the construction in [Anshu, Breuckmann, Nirkhe, STOC 2023]. We define a fermionic analogue of the class quantum PCP and discuss its relation with the qubit version.

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