publications

preprint

2025

1. Experimental observation of multimode quantum phase transitions in a superconducting Bose-Hubbard simulator

   Claudia Castillo-Moreno, Théo Sépulcre, Timo Hillmann, Kazi Rafsanjani Amin, Mikael Kervinen, and Simone Gasparinetti

   Aug 2025

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   The study of phase transitions and critical phenomena arising in quantum driven-dissipative systems, and whether a correspondence can be drawn to their equilibrium counterparts, is a pressing question in contemporary physics. The development of large-scale superconducting circuits provides an experimental platform for these theoretical models. We report an experimental study of a multi-mode dissipative first-order phase transition in a 1D Bose-Hubbard chain consisting of 21 superconducting resonators. This phase transition manifests itself as a simultaneous frequency jump in all resonator modes as the frequency or power of a pump tone is swept. By measuring the system’s emission spectrum through the transition, we characterize the dim-to-bright phase transition and construct the full phase diagram. We further perform time-dependent measurements of the switching between the two phases in the transition region, from which we corroborate the transition line and extract transition times ranging from a few ms up to 143~s. Our model, based on single-mode mean-field theory and cross-Kerr interactions, captures the features at moderate pump powers and quantitatively reproduces the transition line. Our results open a new window into non-equilibrium quantum many-body physics and mark a step toward realizing and understanding dissipative phase transitions in the thermodynamic limit using superconducting quantum circuits.

2. Lattice surgery with Bell measurements: Modular fault-tolerant quantum computation at low entanglement cost

   Trond Hjerpekjøn Haug, Timo Hillmann, Anton Frisk Kockum, and Raphaël Van Laer

   Oct 2025

   Abs Website

   Modular architectures are a promising approach to scaling quantum computers to fault tolerance. Small, low-noise quantum processors connected through relatively noisy quantum links are capable of fault-tolerant operation as long as the noise can be confined to the interface. Finding protocols that implement the quantum links between modules as efficiently as possible is essential because inter-module entanglement is challenging to produce at a similar rate and fidelity as local entanglement. We introduce a protocol for lattice surgery on surface codes in which all non-local operations are Bell measurements. The protocol simultaneously confines the link noise and requires only half as many module-crossing gates as previously proposed protocols. To mitigate distance-reducing hook errors, we introduce a strategy of alternating the gate sequence between rounds of syndrome measurement, which prevents multiple hooks from simultaneously aligning with a logical operator in the code. We evaluate our protocol’s performance when two logical qubits on separate modules are prepared in a logical Bell state. Circuit-level simulations under depolarizing noise show that the logical error suppression for a given entanglement rate between modules is consistently stronger compared to the best-performing alternative protocols for a wide range of link noise, with a typical 40% entanglement resource saving for a constant logical error rate. Our approach to protocol design is applicable to any quantum circuit that must be divided across processor modules and can therefore guide development of resource-efficient modular quantum computation beyond the surface code.

2024

1. High-threshold, low-overhead and single-shot decodable fault-tolerant quantum memory

   Thomas R. Scruby, Timo Hillmann, and Joschka Roffe

   Jun 2024

   Abs Website

   We present a new family of quantum low-density parity-check codes, which we call radial codes, obtained from the lifted product of a specific subset of classical quasi-cyclic codes. The codes are defined using a pair of integers \(r,s) and have parameters \[\![2r^2s,2(r-1)^2,\leq2s]\!] with numerical studies suggesting average-case distance linear in \s\. In simulations of circuit-level noise, we observe comparable error suppression to surface codes of similar distance while using approximately five times fewer physical qubits. This is true even when radial codes are decoded using a single-shot approach, which can allow for faster logical clock speeds and reduced decoding complexity. We describe an intuitive visual representation, canonical basis of logical operators and optimal-length stabiliser measurement circuits for these codes, and argue that their error correction capabilities, tunable parameters and small size make them promising candidates for implementation on near-term quantum devices.

Journal Articles

2026

1. Performance of rotation-symmetric bosonic codes in the presence of non-Markovian effects induced by random telegraph noise

   Adithi Udupa, Timo Hillmann, Rabsan Galib Ahmed, Andrea Smirne, and Giulia Ferrini

   Phys. Rev. Res., Apr 2026

   Abs Website

   Decoherence in quantum devices, such as qubits and resonators, is often caused by bistable fluctuators modeled as random telegraph noise (RTN), leading to significant dephasing. We analyze the impact of individual and multiple fluctuators on a bosonic mode in continuous variable systems, identifying non-Markovian behavior governed by two timescales: the switching rate (ξ) and the coupling strength (ν) of the fluctuator. Using the Breuer-Laine-Piilo trace-distance measure, we characterize non-Markovianity for both Gaussian and non-Gaussian states, revealing that for rotation-symmetric bosonic (RSB) codes, known for their error-correction advantages, the measure grows linearly with code symmetry and can become unbounded. We evaluate the performance of these RSB codes under simultaneous loss and RTN dephasing. For a teleportation-based Knill error-correction circuit, the codes perform robustly in the Markovian limit. In the non-Markovian regime, the performance depends nontrivially on the time at which the error correction is performed. The average gate fidelity of the error-corrected state in this case exhibits oscillations as a function of time due to the oscillatory nature of the dephasing function of the RTN noise; however, for most of the parameter ranges, the values stay beyond the breakeven point. Extending to multiple fluctuators that produce 1/f noise, we observe that non-Markovianity decays with increasing fluctuator count, while the performance of RSB codes remains effective with increasing number of fluctuators.

2025

1. Scaling and networking a modular photonic quantum computer

   H. Aghaee Rad, and  others

   Nature, Feb 2025

   Abs Website

   Photonics offers a promising platform for quantum computing1–4, owing to the availability of chip integration for mass-manufacturable modules, fibre optics for networking and room-temperature operation of most components. However, experimental demonstrations are needed of complete integrated systems comprising all basic functionalities for universal and fault-tolerant operation5. Here we construct a (sub-performant) scale model of a quantum computer using 35 photonic chips to demonstrate its functionality and feasibility. This combines all the primitive components as discrete, scalable rack-deployed modules networked over fibre-optic interconnects, including 84 squeezers6 and 36 photon-number-resolving detectors furnishing 12 physical qubit modes at each clock cycle. We use this machine, which we name Aurora, to synthesize a cluster state7 entangled across separate chips with 86.4 billion modes, and demonstrate its capability of implementing the foliated distance-2 repetition code with real-time decoding. The key building blocks needed for universality and fault tolerance are demonstrated: heralded synthesis of single-temporal-mode non-Gaussian resource states, real-time multiplexing actuated on photon-number-resolving detection, spatiotemporal cluster-state formation with fibre buffers, and adaptive measurements implemented using chip-integrated homodyne detectors with real-time single-clock-cycle feedforward. We also present a detailed analysis of our architecture’s tolerances for optical loss, which is the dominant and most challenging hurdle to crossing the fault-tolerant threshold. This work lays out the path to cross the fault-tolerant threshold and scale photonic quantum computers to the point of addressing useful applications.

2. Localized statistics decoding for quantum low-density parity-check codes

   Timo Hillmann, Lucas Berent, Armanda O. Quintavalle, Jens Eisert, Robert Wille, and Joschka Roffe

   Nat Commun, Sep 2025

   Abs Website

   Quantum low-density parity-check codes are a promising candidate for fault-tolerant quantum computing with considerably reduced overhead compared to the surface code. However, the lack of a practical decoding algorithm remains a barrier to their implementation. In this work, we introduce localized statistics decoding, a reliability-guided inversion decoder that is highly parallelizable and applicable to arbitrary quantum low-density parity-check codes. Our approach employs a parallel matrix factorization strategy, which we call on-the-fly elimination, to identify, validate, and solve local decoding regions on the decoding graph. Through numerical simulations, we show that localized statistics decoding matches the performance of state-of-the-art decoders while reducing the runtime complexity for operation in the sub-threshold regime. Importantly, our decoder is more amenable to implementation on specialized hardware, positioning it as a promising candidate for decoding real-time syndromes from experiments.

3. Single-shot and measurement-based quantum error correction via fault complexes

   Timo Hillmann, Guillaume Dauphinais, Ilan Tzitrin, and Michael Vasmer

   Phys. Rev. A, Oct 2025

   Website
4. Linear-Optical Quantum Computation with Arbitrary Error-Correcting Codes

   Blayney W. Walshe, Ben Q. Baragiola, Hugo Ferretti, José Gefaell, Michael Vasmer, Ryohei Weil, and 8 more authors

   Phys. Rev. Lett., Mar 2025

   Abs Website

   High-rate quantum error-correcting codes mitigate the imposing scale of fault-tolerant quantum computers but require efficient generation of nonlocal, many-body entanglement. We provide a linear-optical architecture with these properties, compatible with arbitrary codes and Gottesman-Kitaev-Preskill qubits on generic lattices, and featuring a natural way to leverage physical noise bias. Simulations of hyperbolic surface codes and bivariate bicycle codes, promising families of quantum low-density parity-check codes, reveal a threshold comparable to the 2D surface code with substantially better encoding rates.

2024

1. Analog Information Decoding of Bosonic Quantum Low-Density Parity-Check Codes

   Lucas Berent, Timo Hillmann, Jens Eisert, Robert Wille, and Joschka Roffe

   PRX Quantum, May 2024

   Abs Website

   Quantum error correction is crucial for scalable quantum information-processing applications. Traditional discrete-variable quantum codes that use multiple two-level systems to encode logical information can be hardware intensive. An alternative approach is provided by bosonic codes, which use the infinite-dimensional Hilbert space of harmonic oscillators to encode quantum information. Two promising features of bosonic codes are that syndrome measurements are natively analog and that they can be concatenated with discrete-variable codes. In this work, we propose novel decoding methods that explicitly exploit the analog syndrome information obtained from the bosonic qubit readout in a concatenated architecture. Our methods are versatile and can be generally applied to any bosonic code concatenated with a quantum low-density parity-check (QLDPC) code. Furthermore, we introduce the concept of quasi-single shot protocols as a novel approach that significantly reduces the number of repeated syndrome measurements required when decoding under phenomenological noise. To realize the protocol, we present the first implementation of time-domain decoding with the overlapping window method for general QLDPC codes and a novel analog single-shot decoding method. Our results lay the foundation for general decoding algorithms using analog information and demonstrate promising results in the direction of fault-tolerant quantum computation with concatenated bosonic-QLDPC codes.

2. Universal control of a bosonic mode via drive-activated native cubic interactions

   Axel M. Eriksson, Théo Sépulcre, Mikael Kervinen, Timo Hillmann, Marina Kudra, Simon Dupouy, and 6 more authors

   Nat Commun, Mar 2024

   Abs Website

   Linear bosonic modes offer a hardware-efficient alternative for quantum information processing but require access to some nonlinearity for universal control. The lack of nonlinearity in photonics has led to encoded measurement-based quantum computing, which relies on linear operations but requires access to resourceful (’nonlinear’) quantum states, such as cubic phase states. In contrast, superconducting microwave circuits offer engineerable nonlinearities but suffer from static Kerr nonlinearity. Here, we demonstrate universal control of a bosonic mode composed of a superconducting nonlinear asymmetric inductive element (SNAIL) resonator, enabled by native nonlinearities in the SNAIL element. We suppress static nonlinearities by operating the SNAIL in the vicinity of its Kerr-free point and dynamically activate nonlinearities up to third order by fast flux pulses. We experimentally realize a universal set of generalized squeezing operations, as well as the cubic phase gate, and exploit them to deterministically prepare a cubic phase state in 60 ns. Our results initiate the experimental field of polynomial quantum computing, in the continuous-variables notion originally introduced by Lloyd and Braunstein.

2023

1. Quantum error correction with dissipatively stabilized squeezed-cat qubits

   Timo Hillmann, and Fernando Quijandría

   Phys. Rev. A, Mar 2023

   Abs Website

   Noise-biased qubits are a promising route toward significantly reducing the hardware overhead associated with quantum error correction. The squeezed-cat code, a nonlocal encoding in phase space based on squeezed coherent states, is an example of a noise-biased (bosonic) qubit with exponential error bias. Here we propose and analyze the error correction performance of a dissipatively stabilized squeezed-cat qubit. We find that for moderate squeezing the bit-flip error rate gets significantly reduced in comparison with the ordinary cat qubit while leaving the phase-flip rate unchanged. Additionally, we find that the squeezing enables faster and higher-fidelity gates.

2. Resolving Fock states near the Kerr-free point of a superconducting resonator

   Yong Lu, Marina Kudra, Timo Hillmann, Jiaying Yang, Hang-Xi Li, Fernando Quijandría, and 1 more author

   npj Quantum Inf, Nov 2023

   Abs Website

   We have designed a tunable nonlinear resonator terminated by a SNAIL (Superconducting Nonlinear Asymmetric Inductive eLement). Such a device possesses a Kerr-free point in which the external magnetic flux allows to suppress the Kerr interaction. We have excited photons near this Kerr-free point and characterized the device using a transmon qubit. The excitation spectrum of the qubit allows to observe photon-number-dependent frequency shifts about nine times larger than the qubit linewidth. Our study demonstrates a compact integrated platform for continuous-variable quantum processing that combines large couplings, considerable relaxation times and excellent control over the photon mode structure in the microwave domain.

2022

1. Designing Kerr Interactions for Quantum Information Processing via Counterrotating Terms of Asymmetric Josephson-Junction Loops

   Timo Hillmann, and Fernando Quijandría

   Phys. Rev. Applied, Jun 2022

   Abs Website

   Continuous-variable systems realized in high-coherence microwave cavities are a promising platform for quantum information processing. While strong dynamic nonlinear interactions are desired to implement fast and high-fidelity quantum operations, static cavity nonlinearities typically limit the performance of bosonic quantum error-correcting codes. Here we study theoretical models of nonlinear oscillators describing superconducting quantum circuits with asymmetric Josephson-junction loops. Treating the nonlinearity as a perturbation, we derive effective Hamiltonians using the Schrieffer-Wolff transformation. We support our analytical results by numerical experiments and show that the effective Kerr-type couplings can be canceled by an interplay of higher-order nonlinearities. This can be better understood in a simplified model supporting only cubic and quartic nonlinearities. Our results show that a cubic interaction allows an increase in the effective rates of both linear and nonlinear operations without an increase in the undesired anharmonicity of an oscillator which is crucial for many bosonic encodings.

2. Performance of Teleportation-Based Error-Correction Circuits for Bosonic Codes with Noisy Measurements

   Timo Hillmann, Fernando Quijandría, Arne L. Grimsmo, and Giulia Ferrini

   PRX Quantum, May 2022

   Abs Website

   Bosonic quantum error-correcting codes offer a viable direction towards reducing the hardware overhead required for fault-tolerant quantum information processing. A broad class of bosonic codes, namely rotation-symmetric codes, can be characterized by their phase-space rotation symmetry. However, their performance has been examined to date only within an idealistic noise model. Here, we further analyze the error-correction capabilities of rotation-symmetric codes using a teleportation-based error-correction circuit. To this end, we numerically compute the average gate fidelity, including measurement errors into the noise model of the data qubit. Focusing on physical measurement models, we assess the performance of heterodyne and adaptive homodyne detection in comparison to the previously studied canonical phase measurement. This setting allows us to shed light on the role of different currently available measurement schemes when decoding the encoded information. We find that with the currently achievable measurement efficiencies in microwave optics, bosonic rotation codes undergo a substantial decrease in their break-even potential. In addition, we perform a detailed analysis of Gottesman-Kitaev-Preskill (GKP) codes using a similar error-correction circuit that allows us to analyze the effect of realistic measurement models on different codes. In comparison to RSB codes, we find for GKP codes an even greater reduction in performance together with a vulnerability to photon-number dephasing. Our results show that highly efficient measurement protocols constitute a crucial building block towards error-corrected quantum information processing with bosonic continuous-variable systems.

2020

1. Universal Gate Set for Continuous-Variable Quantum Computation with Microwave Circuits

   Timo Hillmann, Fernando Quijandría, Göran Johansson, Alessandro Ferraro, Simone Gasparinetti, and Giulia Ferrini

   Phys. Rev. Lett., Oct 2020

   Abs Website

   We provide an explicit construction of a universal gate set for continuous-variable quantum computation with microwave circuits. Such a universal set has been first proposed in quantum-optical setups, but its experimental implementation has remained elusive in that domain due to the difficulties in engineering strong nonlinearities. Here, we show that a realistic three-wave mixing microwave architecture based on the superconducting nonlinear asymmetric inductive element [Frattini et al., Appl. Phys. Lett. 110, 222603 (2017)] allows us to overcome this difficulty. As an application, we show that this architecture allows for the generation of a cubic phase state with an experimentally feasible procedure. This work highlights a practical advantage of microwave circuits with respect to optical systems for the purpose of engineering non-Gaussian states and opens the quest for continuous-variable algorithms based on few repetitions of elementary gates from the continuous-variable universal set.

patent

2024

1. Generation of Gottesman-Kitaev-Preskill (GKP) States Based on Multi-Peak Quantum States of Light

   Rafael N. Alexander, Joseph Eli Bourassa, Jacob Hastrup, and Timo Hillmann

   Oct 2024

Theses

2025

1. Bosonic quantum computing with near-term devices and beyond

   Timo Hillmann

   Chalmers University of Technology, Oct 2025

   Abs Website

   This thesis investigates scalable strategies for fault-tolerant quantum computation by developing and analyzing bosonic quantum codes, quantum low-density parity-check (LDPC) codes, and decoding protocols that aim to unify bosonic and discrete-variable quantum error correction.In the continuous-variable regime, we explore the use of native nonlinearities in superconducting microwave circuits to realize a universal gate set for continuous-variable quantum computing, including the deterministic generation of a cubic phase state. Separately, we propose and analyze the dissipatively squeezed cat qubit, a noise-biased bosonic encoding that offers improved error suppression and faster gate implementations compared to standard cat qubits. To evaluate the broader viability of bosonic encodings, we study the performance of rotation-symmetric and Gottesman-Kitaev-Preskill (GKP) codes under realistic noise and measurement models, revealing important trade-offs in measurement-based approaches.Recognizing the need to integrate bosonic codes into larger fault-tolerant architectures, we develop decoding techniques that explicitly leverage analog syndrome information from the readout of bosonic modes. These methods reduce the need for repeated measurements and enable quasi-single-shot decoding in concatenated schemes, forming a bridge between continuous-variable encodings and discrete-variable stabilizer codes.To advance scalable discrete-variable fault tolerance, we introduce localized statistics decoding, a flexible and highly parallelizable decoding framework for general quantum LDPC codes with state-of-the-art accuracy.Based on a novel on-the-fly matrix elimination strategy, this decoder efficiently identifies and resolves local error configurations, enabling low-latency and hardware-friendly implementations. Additionally, we present quantum radial codes, a new family of single-shot quantum LDPC codes constructed from lifted products of classical quasi-cyclic codes. These codes offer low overhead, tunable parameters, and competitive performance under circuit-level noise, making them promising candidates for near-term implementation.Finally, we propose the concept of fault complexes, a homological framework for representing and analyzing faults in dynamic quantum error correction protocols. Extending the role of homology in static CSS codes, fault complexes provide a general language for the design and analysis of fault-tolerant schemes.

2020

1. Superconducting Quantum Circuits and their Application to Continuous-Variable Quantum Information

   Timo Hillmann

   RWTH Aachen, Aug 2020

Miscellaneous

2024

1. Simulation results for "Localized statistics decoding: A parallel decoding algorithm for quantum low-density parity-check codes"

   Timo Hillmann, Lucas Berent, Armanda O. Quinatavalle, Jens Eisert, Robert Wille, and Joschka Roffe

   Jun 2024

   Abs <a href="Simulation results for "Localized statistics decoding" class="btn btn-sm z-depth-0" role="button">Website</a>

   This dataset contains simulations results presented in the paper  "Localized statistics decoding: A parallel decoding algorithm for quantum low-density parity-check codes". The files are in ‘csv‘ file format, with data easily processable using the python library ‘sinter‘.