Session L07: Autonomous QEC and Bosonic Codes

Protecting a bosonic qubit with autonomous quantum error correction I – Theory

Existing demonstrations of quantum error correction are based on an active schedule of measurement and recovery operations which is hardware intensive and incurs additional error overhead. It is theoretically possible to correct quantum errors with dissipation in a continuous and autonomous fashion, without a classical controller. While dissipative confinement of a quantum system to a two-state manifold had been demonstrated, so far it has remained challenging to achieve a dissipation operator that counters the dominant natural errors in order to extend the lifetime of an encoded qubit. Here we present an autonomous error correction scheme for a bosonic qubit in a superconducting cavity, which directly corrects the dominant error channel of the system: single photon loss. In this Part I of the talk, we discuss this dissipative error correction protocol, its design considerations, as well as its expected performance and limitations.   

Protecting a bosonic qubit with autonomous quantum error correction II – Experiment

Existing demonstrations of quantum error correction are based on an active schedule of measurement and recovery operations which is hardware intensive and incurs additional error overhead. It is theoretically possible to correct quantum errors using dissipation in a continuous and autonomous fashion, without a classical controller. While dissipative confinement of a quantum system to a two-state manifold had been demonstrated, so far it has remained challenging to achieve a dissipation operator that counters the dominant natural errors in order to extend the lifetime of an encoded qubit. Here we present an autonomous error correction scheme for a bosonic qubit in a superconducting cavity, which directly corrects the dominant error channel of the system: single photon loss. In this Part II of the talk, we present our circuit QED setup and experimental results.   

All-Gaussian universality and fault tolerance with the Gottesman-Kitaev-Preskill code

The Gottesman-Kitaev-Preskill (GKP) encoding of a qubit within an oscillator is particularly appealing for fault-tolerant quantum computing with bosons because Gaussian operations on encoded Pauli eigenstates enable Clifford quantum computing with error correction. We show that applying GKP error correction to Gaussian input states, such as vacuum, produces distillable magic states, achieving universality without additional non-Gaussian elements. Fault tolerance is possible with sufficient squeezing and low enough external noise. Thus, Gaussian operations are sufficient for fault-tolerant, universal quantum computing given a supply of GKP-encoded Pauli eigenstates.
  

Stabilization of finite-energy Gottesman-Kitaev-Preskill bosonic codes

Due to their large Hilbert space and their high quality factors, microwave cavities are an attractive candidate for the encoding of logical quantum information. One promising choice of encoding in these cavities is the Gottesman-Kitaev-Preskill (GKP) code which allows to protect against small displacements in phase space. However, in their ideal form, GKP codewords contain an infinite amount of energy and, consequently, their experimental implementation cannot be exact. Nevertheless, recent experiments [Flühmann et al., Nature (2019), Campagne-Ibarcq et al., arXiv:1907.12487] have demonstrated that it is possible to obtain long coherence times for GKP logical qubits. In this talk, we investigate improved stabilization strategies tailored specifically for finite-energy GKP codes and study how these protocols perform in a superconducting implementation with realistic parameters.   

Path-independent gates for error-corrected quantum computing: Theory

Universal control of a quantum system can usually not be achieved by direct control of the system. To realize the missing unitary gates for universal control, we can couple an ancilla system with more complete functionality to the logical system and jointly control both systems. However, the ancilla often suffers much stronger noise than the logical system. Here, we propose a general class of quantum gates on the logical system that is path independent (PI) of Markovian ancilla error trajectories, including both ancilla relaxation and dephasing errors. By fixing the initial and final ancilla states, the designed gates can be PI of infinite-order ancilla dephasing errors, finite-order ancilla relaxation errors, and the combination of both. The PI gates can also be made error-transparent to the first-order logical system errors. As an example, we show that the photon-number selective arbitrary phase (SNAP) gates in circuit QED belong to such a class of PI gates. This proposal provides a hardware-efficient approach toward fault-tolerant quantum computation with system-specific error models.   

Path-Independent Gates for Error-Corrected Quantum Computing: Experiment

In future fault-tolerant quantum computers, errors resulting from noise and decoherence must be detected and corrected in real-time. This is particularly important while applying logical gates, which can cause errors to quickly spread throughout the system.
Here, we present an error-corrected construction for a logical gate set [1] enacted by a multilevel transmon ancilla on a cavity-encoded logical qubit. We show that the logical information is maintained by detecting ancilla errors and applying the appropriate corrections to the logical qubit. The error-corrected operation is path-independent of dominant ancilla errors, leading to a sixfold suppression of the gate error with increased energy relaxation, and a fourfold suppression with increased dephasing noise. The results support the viability of hardware-efficient bosonic quantum computation by showing that bosonic qubits can be controlled by error-prone ancillas without inheriting their inferior performance.

[1] P. Reinhold et al., arXiv:1907.12327 (2019).   

Error-transparent operations on a logical qubit protected by quantum error correction

Universal quantum computation is striking for its unprecedented capability in processing information, but its scalability is challenging in practice because of the inevitable environment noise. Although quantum error correction (QEC) techniques have been developed to protect stored quantum information from leading orders of errors, the noise-resilient processing of the QEC-protected quantum information is highly demanded but remains elusive. Here, we demonstrate phase gate operations on a logical qubit encoded in a bosonic oscillator in an error-transparent (ET) manner. The ET gates are extended to the bosonic code and are able to tolerate errors during the gate operations, regardless of the random occurrence time of the error. With precisely designed gate Hamiltonians through photon-number-resolved AC-Stark shifts, the ET condition is fulfilled experimentally. We verify that the ET gates outperform the non-ET gates with a substantial improvement of the gate fidelity after an occurrence of the single-photon-loss error. Our ET gates in the superconducting quantum circuits are readily for extending to multiple encoded qubits and a universal gate set is within reach, paving the way towards fault-tolerant quantum computation.   

High-impedance circuits for parity measurements of cat qubits

Encoding a qubit in the two degenerate steady states of an oscillator—which only exchanges pairs of photons with its environment—can exponentially suppress the bit-flip rate for large phase-space separations. The unsuppressed phase flips of these so-called "cat qubits" correspond to a change in the photon number parity of the oscillator, and they could be corrected using redundant encoding. In such a scheme, errors are detected via measurements of the joint parity between cat qubits, which could be implemented at the Hamiltonian level using effective parity-type couplings. We show that a parity-type Hamiltonian emerges from the conventional Josephson potential in the limit of high oscillator impedance. Here, the high impedance guarantees large fluctuations of the superconducting phase, which translates into large displacements in oscillator phase space. We present the design of a superconducting circuit that effectively realizes the parity-type Hamiltonian, as well as the status of its experimental implementation.   

Experimental implementation of fault-tolerant error syndrome measurement for pair-cat code (1/2)

Stabilized quantum manifolds of a bosonic system can encode error-protected qubits. In particular, a single-mode manifold spanned by cat states can exponentially suppress against phase-flip errors. However, errors due to photon loss cannot be corrected without stopping the stabilization process, using existing microwave superconducting circuit technology. Phase-flip suppression can also be achieved by stabilizing a manifold spanned by pair-cat states, which are superpositions of the two-mode states called Barut-Girardello/pair-coherent states. Moreover, it is now possible to detect, in a fault-tolerant manner, photon-loss errors in either mode, simultaneously with the manifold stabilization, by monitoring the photon-number difference between them. In this talk, we will present an experimental implementation of cavities and superconducting devices that is compatible with such encoding. We will also report on techniques of continuous monitoring of the photon number difference between the modes. Part-one of this two-part presentation will introduce the basic theoretical concepts of pair-cat codes and the design parameters of our experimental implementation.   

Experimental implementation of fault-tolerant error syndrome measurement for pair-cat code (2/2)

Stabilized quantum manifolds of a bosonic system can encode error-protected qubits. In particular, a single-mode manifold spanned by cat states can exponentially suppress against phase-flip errors. However, errors due to photon loss cannot be corrected without stopping the stabilization process, using existing microwave superconducting circuit technology. Phase-flip suppression can also be achieved by stabilizing a manifold spanned by pair-cat states, which are superpositions of the two-mode states called Barut-Girardello/pair-coherent states. Moreover, it is now possible to detect, in a fault-tolerant manner, photon-loss errors in either mode, simultaneously with the manifold stabilization, by monitoring the photon-number difference between them. In this talk, we will present an experimental implementation of cavities and superconducting devices that is compatible with such encoding. We will also report on techniques of continuous monitoring of the photon number difference between the modes. Part-two of this two-part presentation will present our most recent experimental progress.   

Progress on fault-tolerant quantum computing with concatenated bosonic-qubit codes

In this talk I will discuss ongoing work to quantify the performance of bosonic error correcting codes concatenated with conventional qubit codes. There are two questions we are trying to answer: 1. When does using a bosonic code at the ground level of a concatenated scheme outperform using two-level systems? 2. How do different bosonic codes compare to each other. We would like to answer both of these questions in a fault-tolerant setting that includes state preparation and measurement noise, as well as noise during the error correction circuit.   

Fault-Tolerant Bosonic Quantum Error Correction with the Surface-GKP Code

Bosonic quantum error correction is a viable option for realizing error-corrected quantum information processing in continuous-variable bosonic systems. Here, we consider the concatenation of the bosonic Gottesman-Kitaev-Preskill (GKP) code with the surface code, namely, the surface-GKP code. In particular, we thoroughly investigate the performance of the surface-GKP code by assuming realistic GKP states with a finite squeezing and noisy circuit elements due to photon losses. By using a minimum-weight perfect matching decoding algorithm on a 3D space-time graph, we show that fault-tolerant bosonic quantum error correction is possible with the surface-GKP code if the squeezing of the GKP states is higher than 11.2dB in the case where the GKP states are the only noisy elements. We also show that the squeezing threshold changes to 18.6dB when both the GKP states and circuit elements are comparably noisy. At this threshold, each circuit component fails with probability 0.69%. Finally, if the GKP states are noiseless, fault-tolerant quantum error correction with the surface-GKP code is possible if each circuit element fails with probability less than 0.81%.   

Majorana dimer models of holographic quantum error correction

Holographic quantum error-correcting codes have been proposed as toy models describing key aspects of the AdS/CFT correspondence. In this talk, we introduce a versatile framework of Majorana dimers capturing the intersection of stabilizer and Gaussian Majorana states. This picture allows for an efficient contraction with a simple diagrammatic interpretation and is amenable to analytical study of holographic quantum error-correcting codes. Equipped with this framework, we revisit the recently proposed hyperbolic pentagon code (HyPeC) and demonstrate efficient computation of boundary state properties for generic logical bulk input. We show that the dimers characterizing boundary states of the HyPeC follow discrete bulk geodesics. From this geometric picture, properties of entanglement, quantum error correction, and bulk/boundary operator mapping immediately follow, offering a fresh perspective on holography. We also elaborate upon the emergence of the Ryu-Takayanagi formula from our model, which shares many properties of the recent bit thread proposal.
Our work thus elucidates the connection between bulk geometry, entanglement, and quantum error correction in AdS/CFT, and lays the foundation for new models of holography.
We close with an outlook on the boundary symmetries of holographic Majorana dimer models and discuss the connection to the strong disorder renormalization group.