Session N37: Bosonic Codes and New Quantum Error Correction Schemes

Preparation and Stabilization of GKP states with Kerr-Cat Qubit Ancilla

The Gottesman-Kitaev-Preskill (GKP) code is a promising bosonic error correction code that provides protection against small displacement errors of an oscillator. Preparation and stabilization of GKP states have recently been demonstrated in trapped-ion and circuit-QED platforms. The key component in these experiments is a two-level-ancilla-controlled displacement of the high-Q cavity mode. In the circuit-QED implementation, the main factor limiting the fidelity of GKP preparation and stabilization is the propagation of relaxation or T₁ errors from the ancilla to a large, uncorrectable error in the GKP state. Remarkably, the contagious relaxation errors are highly suppressed in a Kerr-cat qubit, potentially making it an excellent two-level ancilla. Here, we characterize the performance of the GKP preparation and stabilization protocol with Kerr-cat ancilla. We consider dominant sources of errors in the system and identify the parameter regimes where the use of a Kerr cat ancilla results in better performance compared to a conventional two-level ancilla.   

Extending the logical lifetime of a stabilized Gottesman-Kitaev-Preskill qubit with reinforcement learning

Gottesman-Kitaev-Preskill (GKP) encoding of a qubit in an oscillator is a promising candidate for quantum error correction (QEC) in bosonic  systems. We introduce a novel QEC protocol for the GKP encoding, and experimentally implement it with a superconducting microwave cavity coupled to a transmon ancilla qubit. The protocol contains 33 tunable parameters which we optimize in-situ using a reinforcement learning agent. We demonstrate that the agent learns interpretable directions in parameter landscape, which would be difficult and time-consuming to discover and optimize for a human experimentalist. The learned QEC protocol for GKP has process fidelity on par with that of the simplest uncorrectable Fock encoding, and yields X & Z logical Pauli operator lifetimes approaching 1 ms. Our experiment clearly demonstrates the advantage of closed-loop optimization for quantum control over the limited model-based tuneup.   

Demonstration of Heralded Optical Attenuation by Zero Photon Subtraction

Photon subtraction is widely employed for optical quantum state engineering with potential applications in quantum communication, and it can be accomplished by conditional measurements of single photons at the outputs of a beam splitter. Interestingly, conditioning on the detection of zero photons in a similar way also transforms optical states. In particular, this so-called "zero photon subtraction" reduces the mean photon number of any superposition of Fock states. Here we experimentally demonstrate this counterintuitive type of attenuation and observe an interesting dependence on the photon number statistics of the input state. These effects are analogous to that of typical single-photon subtraction and provide some additional insight into physical implementations of creation and annihilation operators in quantum optics.   

Automated discovery of autonomous quantum error correction schemes

We can encode a qubit in the energy levels of a quantum system. Relaxation and other dissipation processes lead to decay of the fidelity of this stored information. Is it possible to preserve the quantum information for a longer time by introducing additional drives and dissipation? The existence of autonomous quantum error correcting codes answers this question in the positive. Nonetheless, discovering these codes for a real physical system, i.e., finding the encoding and the associated driving fields and bath couplings, remains a challenge that has required intuition and inspiration to overcome. In this work, we develop and demonstrate a computational approach based on adjoint optimization for discovering autonomous quantum error correcting codes given a description of a physical system. We implement an optimizer that searches for a logical subspace and control parameters to better preserve quantum information. We demonstrate our method on a system of a harmonic oscillator coupled to a lossy qubit, and find that varying the Hamiltonian distance in Fock space -- a proxy for the control hardware complexity -- leads to discovery of different and new error correcting schemes. We discover what we call the \sqrt{3} code, realizable with a Hamiltonian distance d=2, and propose a hardware-efficient implementation based on superconducting circuits.   

Hardware efficient autonomous error correction protocol (Part I)

Quantum error correction (QEC) would be indispensable when scaling up qubits for building a full-fledged quantum computer. Autonomous quantum error correction (AQEC) is a hardware-efficient way to realize QEC without fast digital feedback control. The Very Small Logical Qubit (VSLQ) is one of the AQEC protocols which requires two qutrits and two lossy resonators for protection against relaxation errors [1]. We propose a new scheme to implement the VSLQ Hamiltonian, which we call the star code, that requires only two-photon processes. The star code can be implemented using any linear coupler that can parametrically drive qutrit-qutrit red and blue sidebands, making the code more flexible and easier to realize than the original design. Our simulations show a significant improvement in logical qubit lifetimes and suppression of dephasing noise. We also discuss other constraints that must be addressed while designing the device for successful implementation.

[1] Kapit et al., Phys. Rev. Lett. 116, 150501   

Hardware efficient autonomous error correction protocol (Part II)

A crucial component of our scheme to implement the Very small Logical Qubit (VSLQ) Hamiltonian is an inter-qutrit coupling mechanism capable of generating strong interactions between various two-qutrit levels while maintaining a small cross-Kerr interaction. We discuss our design of a tunable inductive coupler that can produce relevant blue and red sidebands with rates greater than 6 MHz and suppresses ZZ coupling at the operating DC flux point. We also achieve qutrit-resonator blue sideband rates above 1 MHz which is more than sufficient for the VSLQ protocol. We demonstrate our preliminary results of the logical state stabilization and corresponding improvements in the logical relaxation times. Finally, we discuss the challenges behind realizing the full autonomous quantum error correction where six microwave drives are needed to stabilize an unknown logical superposition state.   

Fault-tolerant quantum computing with bosonic qubits

A popular approach for realizing a fault-tolerant quantum computer is to construct logical qubits encoded in a surface code (or its variants) with bare two-level qubits (e.g., superconducting qubits or trapped-ion qubits) as underlying physical qubits. Recently, various alternative approaches have been proposed, such as using bosonic qubits (e.g., GKP qubits and cat qubits) as underlying qubits instead of bare two-level qubits. Bosonic qubits are distinguished from bare two-level qubits as they are themselves protected via bosonic quantum error correction (bosonic QEC). Moreover, bosonic QEC provides bosonic qubits with special structures that are absent in bare two-level qubits. In this talk, I will provide an overview of various proposals for bosonic-qubit architectures and explain how the unique properties of underlying bosonic qubits (e.g., extra analog information for GKP qubits and noise bias for some cat qubits) may be utilized to reduce the hardware resource overhead for building a fault-tolerant quantum computer.   

Encoding qubits in multimode grid states - Part 1

Bosonic error-correction codes leverage the continuous variables of the Hilbert space to design low-error quantum memories. The Gottesman-Kitaev-Preskill (GKP) code is an example of a bosonic code that provides logical encoding of qubit in a phase space lattice of an oscillator mode. Recent experiments in microwave cavities and trapped ions show promising realization of these codes. In order to further improve over single mode codes, one can concatenate such codes with a qubit code. In this talk, we take a different approach, studying codes based on lattices in the 2n-dimensional phase space of n oscillators. We highlight the advantages of this approach with the specific example of a two-mode code based on a hypercubic lattice. Compared to single mode codes, this code consists of shorter stabilizers and longer logical operators which consequently provides increased protection against errors while reducing ancilla-induced faults during stabilizer measurements. We also discuss the gate set for this code, as well as more general relations between lattice symmetries and logical gates.   

Encoding qubits in multimode grid states - Part 2

Bosonic error-correction codes are emerging as an attractive and hardware-efficient alternative to qubit-based encodings. In particular, the Gottesman-Kitaev-Preskill (GKP) grid codes have generated interest due to predicted long logical lifetimes and the fact that Clifford operations can be implemented using Gaussian circuits. Moreover, they have recently been implemented in microwave cavities and in the motion of trapped ions. However, one recurring challenge with practical implementations of bosonic codes, including the GKP code, is that errors in the ancilla used for quantum control of the oscillator modes propagate as logical errors, limiting the information lifetime. In this talk, we show how this limitation can be overcome using multimode GKP codes. First, we will introduce general qubit ancilla-based error-correction circuits to stabilize the code space of multimode grid codes. Then, we show that stabilizing the code space of a two-mode hypercubic code, ancilla errors mostly propagate as correctable errors. Our new error-correction circuits promise the extension of the logical lifetime orders of magnitude beyond the ancilla lifetime.   

Quantifying Qubit Magic with Gottesman-Kitaev-Preskill Encoding

Quantum resource theories are a powerful framework to characterize and quantify relevant quantum phenomena and identify processes that optimize their use for different tasks. In this work [1], we define a resource measure for magic, the sought-after property in most fault-tolerant quantum computers. In contrast to previous literature, our formulation is based on bosonic codes, well-studied tools in continuous-variable quantum computation. With this new methodology, we are able to prove its properties and connect it to the st-norm, a quantity previously only known to be a one-sided magic witness in qubit systems. In particular, we use the Gottesman-Kitaev-Preskill code to represent multi-qubit states and employ the resource theory of Wigner negativity. The analytical expression of our magic measure allows us to extend current analysis limited to small dimensions, easily addressing systems of up to 12 qubits.

Our work opens the way to link other resources in CV with qubit systems and study their implications for discrete-variable properties.

[1] O. Hahn, A. Ferraro, L. Hultquist, G. Ferrini, and L. García-Álvarez, Quantifying qubit magic with gottesman-kitaev-preskill encoding (2021), arXiv:2109.13018 .   

Saving the cat: Approximate quantum error correction using squeezed Schrödinger cat states

A promising road to quantum error correction is that of bosonic codes, whereby a logical quantum bit is robustly encoded in well-chosen states of a quantum harmonic oscillator. Schrödinger cat codes in particular can correct phase-flip errors in the limit of large displacement of the boson field, while they are still vulnerable to bit-flip errors induced by photon loss. Other bosonic codes - such as the four-legged cat code, the binomial code, or the GKP code - have been studied. A bosonic code that can correct both bit-flip and phase-flip errors within an experimentally viable and scalable scheme is still however a challenge. Here we show that adding squeezing as a resource, squeezed-cat states allow a partial correction of the bit-flip error, therefore suppressing the logical error rate, while improving the protection against phase-flip errors. We develop a full parity-check and recovery protocol that is suitable to be implemented on currently available superconducting architectures. With parameters typical of the experimental state-of-the-art, the code can suppress the logical bit-flip error rate by more than one order of magnitude, already with a moderate amount of squeezing.   

Constructing resource theories of nonclassicality for continuous- and discrete-variable hybrid systems

Most of the physical systems used for quantum information processing consist of harmonic oscillators, described by continuous variables (CV), and effectively finite dimensional systems, described by discrete variables (DV). We propose a method of constructing a resource theory of nonclassicality for such a hybrid CV+DV system from that for a CV system. A monotone measure for the hybrid system can be obtained by performing an appropriate measurement on the DV system and then taking the average of the monotone measure for the conditional state of the CV system over measurement outcomes. As an example, we analyze the metrological power of a partially decohered Schrödinger's cat-like state. Our construction provides a general method of generalizing a convex resource theory defined on a subsystem to an extended system.