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Bosonic quantum error correction codes in superconducting quantum circuits

W. Cai, Y. Ma, W. Wang, C. -L. Zou, L. Sun

arXiv:2010.08699v1quant-ph

TL;DR

Quantum information is vulnerable to noise, while qubit-based QEC approaches require scaling physical-qubit counts and incur large resource overheads. This review synthesizes progress in superconducting bosonic codes, reporting cat-code break-even and control demonstrations while noting that universal fault-tolerant processing remains unavailable.

  • Problem

    Qubit-based QEC approaches require scaling physical-qubit counts and incur large resource overheads, complicating QEC for practical quantum information processing.

    Source: §II · p. 3

  • Method

    The article reviews bosonic QEC development in superconducting circuits and discusses near-term applications and future research directions.

    Source: §VI · pp. 18–19

  • Results

    Cat codes were the first bosonic codes to surpass break-even, with universal logical control and a cPhase gate also demonstrated.

    Source: A. Cat codes · p. 7

  • Takeaways & Limitations

    Bosonic QEC is described as beneficial for protecting information from temporal, propagation-loss, and gate errors, allowing longer storage, propagation distance, and circuit depth.

    Source: §VI · p. 18

    Universal fault-tolerant quantum information processing is not available yet.

    Source: §VI · p. 18

Abstract

from arXiv · show

Quantum information is vulnerable to environmental noise and experimental imperfections, hindering the reliability of practical quantum information processors. Therefore, quantum error correction (QEC) that can protect quantum information against noise is vital for universal and scalable quantum computation. Among many different experimental platforms, superconducting quantum circuits and bosonic encodings in superconducting microwave modes are appealing for their unprecedented potential in QEC. During the last few years, bosonic QEC is demonstrated to reach the break-even point, i.e. the lifetime of a logical qubit is enhanced to exceed that of any individual components composing the experimental system. Beyond that, universal gate sets and fault-tolerant operations on the bosonic codes are also realized, pushing quantum information processing towards the QEC era. In this article, we review the recent progress of the bosonic codes, including the Gottesman-Kitaev-Preskill codes, cat codes, and binomial codes, and discuss the opportunities of bosonic codes in various quantum applications, ranging from fault-tolerant quantum computation to quantum metrology. We also summarize the challenges associated with the bosonic codes and provide an outlook for the potential research directions in the long terms.

I. INTRODUCTION

This review examines bosonic quantum error correction in superconducting circuits, from the motivation and physical advantages of bosonic modes to code families, enabling techniques, applications, and open challenges.

I. INTRODUCTION

  • Practical quantum computers capable of large circuit depth call for QEC-protected logical qubits, expected to use most resources, and a logical qubit with longer coherence than its physical components remains a challenging goal.Qubit-based QEC demonstrations are extremely challenging because of physical-resource overhead and difficulties scaling up qubit numbers; encoding and gate operations on encoded qubits remain elusive [13].Source: §I · p. 1
  • Bosonic modes offer an alternative QEC architecture: their large Hilbert space supports encoding by extending excitation numbers, and they can be scalable, network-friendly physical systems.Bosonic QEC can use a single monitored error syndrome, reducing hardware requirements; modes also support quantum computation, communication, simulation, and metrology.Source: §I · p. 1, Figure 1 · p. 2
  • Superconducting circuit QED is a prominent platform for bosonic QEC, with high-control microwave cavities and demonstrated QEC near or beyond break-even, logical-qubit operations, and fault-tolerant operations.Three-dimensional cavities have single-photon lifetimes up to 1–10 ms; the cited demonstrations include QEC, logical-qubit operations, and fault-tolerant operations.Source: §I · pp. 1–2
  • The review covers GKP, cat, and binomial codes, the techniques for code realization and universal control, and proof-of-principle applications across computation, communication, simulation, and metrology.It concludes by discussing future directions and challenges.Source: §I · p. 2

II. BASICS OF QEC

QEC protects superposed logical information by diagnosing errors without directly measuring codewords and recovering from distinguishable error spaces. Bosonic codes use a single mode’s large Hilbert space for redundant encoding, with increased error rates as the cost of adding photons.

II. BASICS OF QEC

  • QEC stores a superposition in an expanded code space and uses non-destructive syndrome measurements to distinguish error spaces before recovery.The measurements diagnose errors without directly measuring the encoded codewords.Source: §II · p. 3
  • The QEC condition characterizes protection against a given error set; logical states must remain indistinguishable under errors, with distinct error spaces orthogonal.Otherwise, the environment could distinguish the logical basis states and induce uncorrectable deformation errors.Source: §II · p. 3
  • Multi-qubit QEC faces resource overhead and growing error-channel complexity, while surface codes tolerate error rates of about 1% using large two-dimensional qubit arrays [9].Concatenated codes require extremely low gate-error thresholds; extending lifetimes and implementing logical operations remain difficult as qubit counts grow.Source: §II · p. 3
  • A bosonic mode’s infinite Fock-state Hilbert space enables redundant encoding in one degree of freedom, restricting associated errors compared with multi-qubit encoding.Superconducting microwave cavities can have lifetimes up to 1–10 ms; a coupled nonlinear element, typically a two-level qubit, enables control.Source: §II · pp. 3–5
  • For single-photon-loss errors, photon-number parity is the syndrome to monitor, and logical basis states should have equal average photon numbers.Equal averages prevent the environment from distinguishing the logical states from a photon-loss event.Source: §II · p. 5
  • Adding photons for redundancy introduces no new error type, but increases the error rate by a factor of n, where n is the codewords’ average photon number.The resulting increase in error rate is a cost of redundant encoding, which good logical control and QEC may compensate for.Source: §II · p. 5

III. QEC BASED ON BOSONIC CODES

This section reviews theoretical and experimental progress in three single-mode bosonic codes—GKP, cat, and binomial—and summarizes their experimental achievements.

III. QEC BASED ON BOSONIC CODES

  • The section focuses on three bosonic codes based on a single mode: GKP, cat, and binomial codes.It discusses related theories and recent experimental progress.Source: §III · p. 5
  • Figure 3 summarizes recent theoretical and experimental progress in bosonic codes.Source: §III · p. 5
  • Experimental achievements for GKP, cat, and binomial codes are summarized in Figure 4 and Table I.Source: §III · p. 5

A. Cat codes

Cat codes encode information in coherent-state phases and use additional components to create redundancy against photon loss. Their variants support error detection, stabilization, fault-tolerant gates, and noise-biased cat qubits.

Four-component cat codes

  • Four-component cat codes use two dimensions for encoding and two for error detection, enabling single-photon-loss correction with photon parity as a quantum-nondemolition syndrome at sufficiently large α.The code and error spaces have even and odd photon-number parity, respectively; the basis states are not exactly orthogonal unless α is sufficiently large.Source: A. Cat codes · p. 5
  • A single-photon loss moves the state between code and error spaces with a π/2 logical Z phase shift, allowing errors to be tracked and recovery deferred until the process ends.The encoded information is not fully recovered until the fourth photon loss, avoiding imperfect recovery operations after each detection.Source: A. Cat codes · pp. 5–6

Experiments

  • Figure 4 presents Wigner functions of logical-qubit basis states in code and error spaces, alongside experimental demonstrations of QEC using cat, binomial, and GKP bosonic codes, adapted from Refs. [12, 35, 36].Source: Figure 4 · p. 6

Cat-code limitations

  • Without quantum jumps, cat states shrink toward vacuum, so energy must be repumped before coherent-state overlap causes non-orthogonality that unitary operations cannot fully correct.The basis states have the same average photon number, but are not exactly orthogonal unless α is sufficiently large.Source: A. Cat codes · pp. 5–6

Stabilization and gates

  • Four-photon dissipation or an engineered Hamiltonian can stabilize cat states; stabilized codes support X rotations and two-qubit entangling gates, though strong four-photon drives remain challenging.The Hamiltonian uses a four-photon drive and high-order Kerr nonlinearity to confine the logical states to the cat manifold.Source: A. Cat codes · pp. 6–7

Larger and multimode cat codes

  • An n-component cat code corrects photon-loss errors up to order n/2 − 1, but requires increased average photon number for the orthogonality condition; pair-cat codes protect against arbitrary losses in one mode if the other is error-free.Multimode cat codes can protect against photon loss through active syndrome measurement or an autonomous procedure.Source: A. Cat codes · p. 7

Cat qubits

  • In experiments, stabilized cat qubits showed biased noise and coherent X rotations, while their bit-flip error is exponentially suppressed with photon number and phase-flip error grows linearly.Their two-state manifold is too small to correct photon loss, but stabilization and phase-space separation suppress local-displacement noise.Source: A. Cat codes · p. 7
  • Stabilized cat qubits can support universal bias-preserving gates, fault-tolerant syndrome detection as ancillas, and surface codes tailored to biased noise.The source identifies biased-noise cat qubits as resources for fault-tolerant quantum computation.Source: A. Cat codes · p. 7

B. Binomial codes

Binomial codes use truncated-Fock superpositions to correct selected orders of photon loss, gain, and dephasing. Their spaced Fock support enables syndrome-based recovery, while the lowest-order code illustrates how a bosonic encoding can use fewer physical resources than a four-qubit code.

B. Binomial codes

  • Binomial codes superpose truncated Fock states with binomial weights and target polynomial photon-loss, photon-gain, and dephasing errors through orders L, G, and D.The error set contains I, a through a^L, a† through (a†)^G, and n through n^D.Source: B. Binomial codes · p. 7
  • The spacing is S = L + G, N = max{L,G,2D}, p ranges from 0 to N + 1, and the maximum Fock number is (N + 1)(S + 1).The passage also describes N + 1 as the binomial coefficient.Source: B. Binomial codes · p. 7
  • Distinct Fock-state series make the two code basis states orthogonal with equal average photon numbers, satisfying the QEC conditions and enabling syndrome detection by photon number modulo S + 1.Because the code space and error spaces are orthogonal, binomial codes allow explicit unitary energy repumping; truncated support may also make unitary operations more convenient than in cat codes.Source: B. Binomial codes · p. 8
  • The lowest-order binomial code corrects single-photon loss with average photon number two, below the typical cat code, but its error states have unequal average photon numbers and require immediate unitary recovery.Without prompt recovery, quantum information can be corrupted.Source: B. Binomial codes · p. 8
  • With the same average excitation of two, the lowest-order binomial code occupies five oscillator levels and needs one error syndrome, versus 16 dimensions and three syndromes for the four-qubit code.The comparison illustrates the bosonic code's physical-resource savings for correcting single amplitude-damping errors.Source: B. Binomial codes · p. 8
  • A higher-order code can target {I, a, a^2, n}; spacing three also lets it correct {I, a, a†, n}, and binomial and cat codes asymptotically approach each other at large average photon number.Photon gain and two-photon loss have the same change in photon number modulo three, the code's generalized-parity spacing.Source: B. Binomial codes · p. 9
  • Even without a detected photon loss, non-unitary no-error evolution distorts binomial code states and must be corrected; a two-mode version can mitigate this backaction.The two-mode version keeps the same spacing and total excitation number while distributing excitations across modes.Source: B. Binomial codes · p. 9
  • Experiments demonstrated repetitive real-time-feedback QEC with lifetime nearly beating break-even, a high-fidelity universal logical gate set, and cPhase and CNOT operations between binomial logical qubits.Reported operations include a geometric cPhase, a teleported CNOT, and a CNOT with a binomially encoded target.Source: Figure 5 · p. 8, B. Binomial codes · p. 9

C. GKP codes

GKP codes encode qudits in oscillator grid states and correct small phase-space shifts using stabilizer measurements. Their error protection, gate operations, physical approximations, experimental demonstrations, and implementation challenges are reviewed.

C. GKP codes

  • First proposed in 2001, GKP codes encode a qudit in an oscillator; typical qubit basis states are grid states spaced by 2√π and stabilized by commuting operators.Their corresponding Pauli operators are displacements by half the grid spacing.Source: C. GKP codes · p. 9
  • Ideal GKP codewords require infinite photon number and are unphysical; approximate states use finite squeezing and a Gaussian envelope.The approximate codewords replace delta functions with finitely squeezed Gaussians and the uniform superposition profile with an envelope centered at q = 0.Source: C. GKP codes · p. 9
  • GKP codes outperform cat, binomial, and numerically optimized bosonic codes for most photon-loss rates, while hexagonal codes provide symmetric error resistance.Numerical optimization for photon-loss and Gaussian thermal-loss channels converges to the hexagonal GKP code.Source: C. GKP codes · p. 9
  • At equal average photon number, GKP codes occupy a broader photon-number distribution and Hilbert space, increasing Kerr distortion and motivating further optimization [80].Clifford gates are Gaussian operations that preserve small q and p deviations, whereas non-Clifford gates and codeword preparation require challenging non-Gaussian resources.Source: C. GKP codes · p. 10
  • Stabilizer measurements identify correctable shifts when |δq| and |δp| are each below √π/2, enabling minimal-displacement correction of local errors.Photon loss, thermal noise, dephasing, and ancilla-induced nonlinearities can produce local phase-space effects; frequent measurements can detect and correct small shifts.Source: C. GKP codes · p. 10
  • Because GKP codes are difficult to scale by increasing squeezing alone and do not protect against rare large errors, they are often concatenated with stabilizer codes [81].Theoretical surface-GKP threshold estimates require about 10–20 dB of squeezing under different assumptions about error sources [83].Source: C. GKP codes · p. 10
  • Experiments demonstrated GKP-qubit encoding, readout, and control in a trapped ion, and square and hexagonal GKP-code QEC in a superconducting cavity.In the cavity, repeated stabilizer measurements and feedback enabled deterministic state generation from vacuum; continuous QEC extended logical-qubit coherence.Source: C. GKP codes · p. 10

IV. UNIVERSAL QUANTUM CONTROL OF BOSONIC CODES

The section reviews circuit-QED control of bosonic modes using an ancilla, covering universal unitary operations and quantum-channel implementations. It describes experimental progress and the challenges of scaling control to logical gates and open systems.

IV. UNIVERSAL QUANTUM CONTROL OF BOSONIC CODES

  • In circuit QED, ancilla-induced nonlinearity enables bosonic-mode control, while SNAP phase gates preserve photon number and displacements connect adjacent Fock states.Their combination provides universal cavity control; a one-photon Fock state was experimentally generated with this method.Source: §IV · pp. 10–11
  • GRAPE optimizes control pulses through gradient ascent over discretized evolution steps and has been applied to QEC encoding and logical-qubit gates [18].The optimization uses forward and backward propagations of the initial and target density operators to update control parameters.Source: §IV · pp. 11–12
  • A two-cavity SNAP gate deterministically generated a single-photon Bell state by applying a π phase shift to |0⟩⊗|0⟩, while scaling control remains challenging.SNAP and GRAPE can be generalized to logical-qubit gates, but larger Hilbert spaces make pulse optimization and experimental realization difficult.Source: §IV · p. 12
  • Arbitrary bosonic-mode quantum channels can be implemented by repeatedly resetting and measuring an ancilla, then using measurement outcomes for feedforward control.The proposed approach is described as hardware-efficient and applicable to initialization, generalized measurements, open-system simulation, and quantum metrology.Source: §IV · p. 12
  • Experiments demonstrated maximally mixed state preparation and autonomous QEC on binomial and truncated four-component cat codes [18], avoiding real-time feedback.In the binomial-code implementation, a unitary transfers error entropy to the ancilla and returns the logical state from the error space to the code space.Source: §IV · p. 12
  • Intrinsic Pockel and Kerr nonlinearities could enable universal bosonic-mode control by simulating arbitrary Hamiltonians through Trotterization, offering a route suited to continuous-variable encodings.The passage notes that this route avoids approximating a truncated Fock space.Source: §IV · p. 12

V. APPLICATIONS OF BOSONIC CODES

Bosonic codes are positioned for quantum-information storage and transfer, while their large Hilbert spaces offer opportunities in quantum simulation and metrology. The section reviews proof-of-principle progress and future prospects, including fault-tolerant operations and applications of QEC or approximate QEC to near-term noisy platforms.

V. APPLICATIONS OF BOSONIC CODES

  • Bosonic codes are foreseeable tools for storing and transferring quantum information in quantum computation and communication.The passage notes that all ingredients of bosonic codes are available in superconducting quantum circuits.Source: §V · p. 12
  • A factor-of-five suppression of ancilla errors was demonstrated in fault-tolerant quantum error detection using an auxiliary ancilla level [15].Figure 7(a–b) depicts the detection scheme and experimental result.Source: Figure 7 · p. 13
  • A path-independent phase gate, designed to keep ancilla populations independent of the encoded state, significantly improved logical-gate benchmarking [16].Figure 7(c–d) presents the gate principle and benchmarking results.Source: Figure 7 · p. 13
  • Error-transparent gates make evolution deterministic and identical in code and error spaces regardless of error timing; experiments showed improved logical-qubit performance with repetitive and interleaved gates and autonomous QEC [17].Figure 7(e–f) shows the concept and process-fidelity results over time.Source: Figure 7 · p. 13
  • Bosonic codes may reduce noise and system imperfections in estimating certain outputs, without requiring standard codes to satisfy the QEC condition.The section also identifies quantum simulation and metrology as opportunities, while noting that reported results remain preliminary and systematic investigation is needed.Source: §V · p. 13

A. Fault-tolerant quantum computation

Bosonic codes have enabled fault-tolerant operations that address errors during computation, including demonstrations that protect encoded information during gates. These efforts have not yet established a fault-tolerant threshold, and further progress may require higher-order single-mode codes or multi-mode encodings.

A. Fault-tolerant quantum computation

  • Bosonic codes have demonstrated encoding, decoding, error correction, and universal logical gates, while early fault-tolerance experiments correct certain gate-time physical errors rather than demonstrate a threshold.Fault tolerance also requires preparations, detections, gates, and measurements to avoid propagating and accumulating errors.Source: A. Fault-tolerant quantum computation · p. 13
  • Ancilla errors can corrupt encoded information during non-Gaussian detection; redundant ancilla levels suppressed ancilla errors fivefold, while simulations with biased-noise cat ancillas suppressed measurement backaction.The cat-ancilla strategy relies on an interaction commuting with σ_z and a cat qubit whose bit-flip rate is exponentially suppressed with size.Source: A. Fault-tolerant quantum computation · p. 14
  • Path independence requires deterministic encoded evolution despite ancilla errors up to a specified order; a path-independent SNAP phase gate on a three-level ancilla significantly improved gate fidelity [59].Other ancilla outcomes can herald failure without corrupting the encoded system.Source: A. Fault-tolerant quantum computation · p. 14
  • Error-transparent phase gates on the lowest-order binomial code preserved code- and error-space states and improved the QEC-protected logical-state lifetime during gate operation.The error-transparent approach could be generalized to a universal gate set.Source: A. Fault-tolerant quantum computation · p. 14
  • Bosonic codes still lack a fault-tolerant threshold; proposed directions are higher-photon-number single-mode codes that tolerate more errors and multi-mode nonlocal encodings.The ultimate goal is suppressing error rates below a threshold without exponentially scaled physical-resource overhead.Source: A. Fault-tolerant quantum computation · p. 14

B. Quantum communications with bosonic codes

Bosonic codes could support quantum networks by protecting stored and transmitted information from local noise, channel noise, and interface losses. Proof-of-principle network components have been demonstrated, alongside proposals for communication, distributed computing, and sensing.

B. Quantum communications with bosonic codes

  • Proof-of-principle experiments have demonstrated basic quantum-network components using superconducting bosonic codes.Such codes can protect information from local storage noise, wavepacket channel noise, and interface insertion loss caused by impedance mismatch.Source: B. Quantum communications with bosonic… · pp. 14–15
  • Storage-cavity states can be converted to traveling wavepackets, while binomial-code pitch-and-catch detects and corrects photon loss for on-demand entanglement.A coaxial bus has also entangled quantum memories, with even-parity encoding tracking photon-loss events during two-photon interference to improve entanglement fidelity [20].Source: B. Quantum communications with bosonic… · p. 15
  • Deterministic teleportation of a CNOT gate between two bosonically encoded nodes has been demonstrated.Repeaters and teleportation could enable high-fidelity communication over long distances, while shared entanglement could support distributed computation between nodes.Source: B. Quantum communications with bosonic… · pp. 15–16
  • Room-temperature thermal noise limits superconducting bosonic networks; microwave-to-optical transducers could combine microwave encoding with optical transmission over thousands of kilometers.Optical photons are limited by probabilistic gate operations; a theoretical study predicts a high secure key rate for memory-less, one-way long-distance communication with cat codes.Source: B. Quantum communications with bosonic… · p. 16
  • Quantum networks could improve sensing through distributed correlated probes, including higher-precision long-baseline quantum telescopes.Source: B. Quantum communications with bosonic… · p. 16

C. Quantum simulations with bosonic codes

Bosonic modes support analog and digital quantum simulations, including molecular vibrations and topological phases; although prior simulators had not directly used QEC codes, bosonic error correction and mitigation could extend NISQ simulation depth.

C. Quantum simulations with bosonic codes

  • Bosonic modes are indispensable to models including boson sampling, molecular vibration, the quantum Rabi and Bose-Hubbard models, and non-Markovian environments, and can also enable analog simulations.The passage frames quantum simulation as an early NISQ-era application spanning chemistry, optimization, materials engineering, and fundamental physics.Source: C. Quantum simulations with bosonic… · p. 16
  • A transmon coupled to a 3D cavity simulated diatomic-molecule vibronic spectra by mapping qubit levels to electronic states and the cavity mode to vibrational motion.The simulator obtained molecular spectra for equilibrium and non-equilibrium states.Source: C. Quantum simulations with bosonic… · p. 16
  • Two microwave cavities and three transmons represented vibrational modes and their interactions, enabling photoelectron spectra simulations for H2O, O3, NO2, and SO2.The processor used single-shot photon-number detection resolving up to 15 photons.Source: C. Quantum simulations with bosonic… · p. 17
  • A split-step quantum walk digitally simulated spin-orbit-coupled particles on a lattice and measured a topological invariant through interference between components of a cavity Schrödinger cat state.The protocol addresses characterization of topological ordering, whose non-local nature makes direct measurement in solid-state materials challenging.Source: C. Quantum simulations with bosonic… · p. 17
  • Under a coarse estimate, 5% imperfect logical-gate error with 99% error-detection efficiency is suppressed to 1%, while 80% expected fidelity supports circuit depth from 5 to 20 at about 36% success probability.The passage presents these as estimates for combining bosonic error correction and error mitigation, not as demonstrated results.Source: C. Quantum simulations with bosonic… · p. 17
  • Bosonic modes suit high-dimensional digital simulations through hardware efficiency, and QEC techniques could enable deeper circuits alongside error mitigation.The passage notes that QEC codes had not yet been directly used in the previous bosonic quantum simulators.Source: C. Quantum simulations with bosonic… · p. 17

D. Quantum metrology with bosonic codes

Bosonic modes offer routes to quantum-enhanced metrology through high-variance probe states and error-corrected code subspaces. Experiments show improved sensitivity and longer coherence, while QEC-enhanced sensing still faces practical challenges.

D. Quantum metrology with bosonic codes

  • Bosonic codes suit sensing across conventional bosonic systems, while superconducting circuits and their spin or mechanical-resonator hybrids offer a platform for high-performance metrology.Conventional systems have limited capability for nondeterministic quantum state engineering, processing, or detection.Source: D. Quantum metrology with bosonic codes · p. 17
  • Coherent-state estimation is limited by shot noise ∝1/N and coherence-time-limited interaction duration ∝1/Tc, while the Heisenberg limit is ∝1/N.The passage states that exploring a bosonic mode’s large Hilbert space could resolve both limits.Source: D. Quantum metrology with bosonic codes · p. 17
  • Preparing probes with maximum variance of HI can suppress shot noise; single-mode experiments in trapped ions and superconducting circuits demonstrated sensitivity approaching HL scaling.Two-mode interferometers have also used squeezed, number, and Schrödinger cat states.Source: D. Quantum metrology with bosonic codes · p. 17
  • Twice as long a coherence time was measured in a QEC-protected binomial-code Ramsey experiment as without QEC, revealing bosonic codes’ sensing potential.The result used a logical qubit in a Ramsey experiment, a technique widely used for precision measurements.Source: D. Quantum metrology with bosonic codes · p. 17
  • Approximate QEC demonstrated an advantage for bosonic radiometry in a superconducting circuit, although the HNLS condition was not completely satisfied.The HNLS condition is a challenge for constructing an optimal code that achieves Heisenberg-limit scaling.Source: D. Quantum metrology with bosonic codes · p. 17

VI. DISCUSSIONS AND OUTLOOK

Bosonic codes offer hardware efficiency, large Hilbert spaces, and long-distance transfer, but realizing their potential requires further demonstrations and advances in hardware, theory, and quantum control. The review identifies near-term applications and long-term paths toward universal quantum computation across multiple bosonic platforms.

Short-term research

  • Bosonic QEC can protect information from temporal, propagation, and gate errors before universal fault-tolerant processing is available, enabling longer storage, propagation, and circuit depth and motivating NISQ applications.The proposed near-term applications include quantum repeaters, simulation, metrology, and machine learning; experimental metrology results surpassed the standard limit by about 9.1 dB at N = 12.Source: §VI · p. 18

System extensions

  • Bosonic systems should extend to multiple oscillators and other platforms: resonator arrays and multilayer-integrated cavities support scaling, while microwave-to-optical transducers and hybrid phononic modes enable networks, memory, and sensing.The demonstrated transducers avoid MHz-frequency mechanical noise; mechanical modes also offer compact, high-density integration.Source: §VI · pp. 18–19

Material and fabrication

  • Long-term universal computation requires improved superconducting materials and fabrication, better characterization of loss mechanisms, and integration that controls frequency crowding and crosstalk while improving coherence, yield, stability, and robustness.High-frequency qubits and resonators at millimeter wavelengths could reduce costs and thermal background noise.Source: §VI · p. 19

Theory

  • A clear bosonic-code fault-tolerance threshold is lacking; proposed theory directions include higher-energy and higher-dimensional codes, multimode extensions, surface-code concatenation, and distributed topological codes.Hardware-adaptive codes, code interconversion, and applications in metrology, simulation, and quantum networks also need study.Source: §VI · p. 19

Advanced quantum control techniques

  • Limited gate fidelity obstructs higher-order bosonic codes, with losses arising from incoherent processes, incomplete control models, and experimental parameter errors; robust control and fuller open-system models could help.Machine-learning control may improve calibration and algorithm efficiency, while combining moderate Pockel or Kerr nonlinearities with spin-oscillator models may aid universal control at higher photon numbers.Source: §VI · p. 19

Outlook

  • Bosonic modes are universal in nature, and superconducting-circuit demonstrations could extend to optical frequencies, mechanical oscillators, spin waves, trapped ions, NV centers, and optical cavity QED.The review anticipates that bosonic codes will contribute to quantum information technologies in both the short and long term.Source: §VI · p. 19

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