Session A67: Continuous-Variable Quantum Information: Bosonic Codes and Error Correction

Stabilizer subsystem decompositions for single- and multi-mode Gottesman-Kitaev-Preskill codes

The Gottesman-Kitaev-Preskill (GKP) error correcting code encodes a finite dimensional logical space in one or more bosonic modes, and has recently been demostrated in trapped ions and su- perconducting microwave cavities. In this work we introduce a new subsystem decomposition for GKP codes that we call the stabilizer subsystem decomposition, analogous to the usual approach to quantum stabilizer codes. The decomposition has the defining property that a partial trace over the non-logical stabilizer subsystem is equivalent to an ideal decoding of the logical state. We describe how to decompose arbitrary states across the subsystem decomposition using set of transformations that move between the decompositions of different GKP codes. Besides providing a convenient theoretical view on GKP codes, such a decomposition is also of practical use. We use the stabilizer subsystem decomposition to efficiently simulate noise acting on single-mode GKP codes, and in contrast to more conventional Fock basis simulations, we are able to to consider essentially arbitrarily large photon numbers for realistic noise channels such as loss and dephasing.   

Bias-preserving operations in pair-cat code

Fault-tolerant quantum computation with depolarization error often requires demanding error threshold and resource overhead. If the operations can maintain high noise bias -- dominated by dephasing error with small bit-flip error -- we can achieve hardware-efficient fault-tolerant quantum computation with a more favorable error threshold. Distinct from two-level physical systems, multi-level systems (such as harmonic oscillators) can achieve a desirable set of bias-preserving quantum operations while using continuous engineered dissipation or Hamiltonian protection to stabilize to the encoding subspace. For example, cat codes stabilized with driven-dissipation or Kerr nonlinearity can possess a set of biased-preserving gates while continuously correcting bosonic dephasing error. However, cat codes are not compatible with continuous quantum error correction against excitation loss error, because it is challenging to continuously monitor the parity to correct photon loss errors. In this work, we generalize the bias-preserving operations to pair-cat codes, which can be regarded as a multimode generalization of cat codes, to be compatible with continuous quantum error correction against both bosonic loss and dephasing errors. Our results open the door towards hardware-efficient robust quantum information processing with both bias-preserving operations and continuous quantum error correction simultaneously correcting bosonic loss and dephasing errors.

    

Efficient qudit encoding for quantum optics

Linear optics are a promising alternative for the realization of quantum computation and quantum communication protocols due to the long coherence time of photons. Linear optics can implement high dimensional computational units called qudits which provide a natural way to compile d-ary algorithms into quantum gates. This work investigates qudits defined by the possible photon number states of a single photon in d > 2 optical modes. We demonstrate how to construct locally optimal non-deterministic many-qudit gates using linear optics and photon number resolving detectors, and explore the use of qudit cluster states in the context of a d-ary optimization problem. We find that the qudit cluster states require less optical modes and are encoded by a fewer number of entangled photons than the qubit cluster states with similar computational capabilities.   

Linear optics and photodetection achieve near-optimal unambiguous coherent state discrimination

Coherent states of the quantum electromagnetic field, the quantum description of ideal laser light, are prime candidates as information carriers for optical communications. 

A large body of literature exists on their quantum-limited estimation and discrimination. However, very little is known about the practical realizations of receivers for unambiguous state discrimination (USD) of coherent states. Here we fill this gap and outline a theory of USD with receivers that are allowed to employ: passive multimode linear optics, phase-space displacements, auxiliary vacuum modes, and on-off photon detection. Our results indicate that, in some regimes, these currently-available optical components are typically sufficient to achieve near-optimal unambiguous discrimination of multiple, multimode coherent states.   

Error Resilient Gate Designs for Dissipative Cat Qubits

The cat qubit encoding is a promising candidate for quantum error correction in bosonic systems. Its inherent exponential protection of the bit value leaves only the need for the quantum phase value of the qubit to be actively corrected. However, its practical implementation may be limited by low gate fidelities and by the poor scaling of usual gate methods with the relevant experimental parameters. Indeed, the typical Hamiltonians that can be engineered in superconducting circuits are very different from the ones that would implement ideal gates on an encoded cat qubit. Hence, cat qubit gates are often only approximated, e.g. using the Zeno dynamics together with the cat stabilization dynamics. In this talk, we will introduce a new perspective on the errors induced by such approximate gate implementations, and build on it to devise four designs that can reduce gate errors by orders of magnitudes at the cost of moderate experimental overhead. These designs should give a broad overview of errors for dissipative cat qubit gates, and inspire new ideas on how to tackle them.   

Quantum error correction with dissipatively stabilized squeezed cat qubits

Noise-biased qubits are a promising route toward significantly reducing the hardware overhead associated with quantum error correction. The squeezed cat code, a non-local 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 cat qubit while leaving the phase flip rate unchanged. Additionally, we find that the squeezing enables faster and higher-fidelity gates.   

Error-detectable logical 2-qubit gates on bosonic qubits

Bosonic error correction in circuit QED has previously been successful at preserving a quantum memory but realizing two-qubit gates in the logical subspace is an outstanding challenge. In particular, we must engineer these operations without succumbing to errors in the requisite nonlinear resource. We present a continuous family of two-qubit gates that satisfy these requirements for bosonic qubits housed in microwave cavities with transmon ancillae for quantum control. Crucially, we have the ability to detect errors in both the cavities and transmon which is vital to preserving the desirable properties of our Bosonic encodings. Our gates require simple hardware - a microwave-actuated beamsplitter interaction between cavities and a transmon ancilla dispersively coupled to only one of the cavities. We show how to construct arbitrary excitation-preserving two-qubit gates for rotation-symmetric codes such as the cat code and other bosonic codes. With appropriate measurements, we can error-detect transmon decay, dephasing and cavity photon loss.   

Photon-number-selective two-excitation transitions for bosonic quantum error correction

Single-photon loss is the dominating source of error in bosonic quantum error correcting codes using microwave resonators. To correct for this type of error, we can use code words of one defined parity (such as cat states or binomial states). Using an ancilla qubit dispersively coupled to a high quality resonator, we propose to realize a collection of photon-number-selective simultaneous single excitations on both the resonator and the qubit.  These transitions are implemented by driving the system with a comb of microwave tones and serve a double purpose. By applying them, we can simultaneously detect the presence of an error and move back to the code space.

Being number-selective, these drives do not affect the code states so they can be applied whether an error has happened or not. We present the experimental implementation of the drives and the phase control they enable on superpositions of Fock states. When supplemented by qubit reset, the presented sequence is suitable for quantum error correction of against single-photon loss.

    

Composite Pulses in Phase Space: Measurement-Free Gate Teleportation with Hybrid Oscillator-Qubit Systems

The quantum operations required to manipulate states encoded in the phase space of an oscillator can be challenging to implement. Recent experimental work has realized universal control of a quantum system by combining fast single-qubit rotations with high-fidelity echoed conditional displacements (ECD), indicating the potential of achieving universal quantum computation with continuous variable (CV) qubits using two-level ancillary qubits. In this talk, we present a framework consisting of entangling and disentangling gadgets for a hybrid oscillator-qubit system that provides an analytical method for constructing these ECD-based circuits. Our method can be seen as composite pulse sequences in phase space for operations which entangle and disentangle oscillator-qubit states without any measurement. By combining these gadgets with single-qubit rotations, one can achieve measurement-free control over certain CV states. As an example, we introduce a measurement-free gate teleportation scheme that provides universal control on finite-energy GKP qubits.   

Scalable Quantum Computing with Bosonic Qubits

Bosonic or continuous-variable qubits are encoded in non-classical states of electromagnetic fields. Rapid technological advances in quantum system design and control, particularly in circuit quantum electrodynamics, have brought bosonic quantum computing to the forefront in the race for building practical quantum technology. The main motivation behind using such encodings is that they can be made more robust to noise either by design or by performing some active error detection or correction directly on the bosonic mode. In this talk I will review recent theoretical and experimental progress in bosonic qubit technology, and present my outlook on the opportunities for scalable fault-tolerance with it.

 

    

Fault-Tolerant Bell Measurements with the 4-Legged Schr?dinger-Cat Code

Previously, the 4-legged cat code has been used to realize a quantum memory surpassing the breakeven point [1] but we wish to perform a wider set of logical operations using this encoding. We propose a new scheme to perform destructive Bell measurements on pairs of 4-legged cat codes which requires only a beam-splitter coupling between two cavities and dispersively coupled transmon ancillae. Importantly, noise in these Bell measurements is at worst second order in the cavities’ single-photon loss and transmons’ relaxation and dephasing errors. With these Bell measurements, the cat-code can now be fused into a surface code cluster state. Since these are first-order fault tolerant measurements, we may achieve substantially lower logical error rates when concatenated with the surface code, than otherwise possible with unencoded physical qubits.

[1] Ofek et. al. Nature 536, 441–445 (2016)   

An error-detectable logical measurement of two bosonic modes

Encoding information in the long-lived bosonic modes of high-Q superconducting cavities allows one to perform the first layer of quantum error correction in a single device. For example, storing logical information in Schrodinger cat states enables protection against single photon loss. In this encoding, measurements of the joint photon number parity of two cavities (a logical-ZZ measurement) have been performed by coupling them to a common transmon ancilla, but are limited by decay and dephasing errors in the lossier transmon. In this talk, we demonstrate a measurement of the joint photon number parity that relies on a transmon ancilla coupled to just one cavity plus a tunable beamsplitter interaction between cavities, and that detects both ancilla dephasing and decay errors. We further show how this measurement can be used as an error-detectable entangling operation for a variety of bosonic codes.   

Majorization ladder in bosonic Gaussian channels

We show the existence of a majorization ladder in bosonic Gaussian channels, that is, we prove that the channel output resulting from the nth energy eigenstate (Fock state) majorizes the channel output resulting from the (n+1)th energy eigenstate (Fock state). This reflects a remarkable link between the energy at the input of the channel and a disorder relation at its output as captured by majorization theory. This result was previously known in the special cases of a pure-loss channel and quantum-limited amplifier, and we achieve here its nontrivial generalization to any single-mode phase-covariant (or -contravariant) bosonic Gaussian channel. The key to our proof is the explicit construction of a column-stochastic matrix that relates the outputs of the channel for any two subsequent Fock states at its input, which is made possible by exploiting a recently found recurrence relation on multiphoton transition probabilities for Gaussian unitaries [M. G. Jabbour and N. J. Cerf, Phys. Rev. Research 3, 043065 (2021)]. We then discuss possible generalizations and implications of our results.