Session V31: Quantum Computing with Continuous-Variable Systems

Live

An infinite dimensional system such as a quantum harmonic oscillator offers a potentially unbounded Hilbert space for computation, but accessing and manipulating the entire state space requires a physically unrealistic amount of energy. When such a simple harmonic oscillator is coupled to a qubit, for example via a Jaynes-Cummings interaction, it is well known that the Hilbert space can be separated into independently accessible subspaces of constant energy, but the number of subspaces is still infinite. Nevertheless, a closed four-dimensional Hilbert space can be formed from the lowest energy states of the qubit + oscillator. We extend this idea, and show for arbitrary N how a 2(N+1)-dimensional Hilbert space can be constructed, which is closed under a certain set of unitary operations resulting solely from manipulating standard Jaynes-Cummings Hamiltonian terms. Moreover, these unitaries are a universal set for quantum operations on this “qubit-oscillator qudit”. This work suggests that the combination of a qubit and a bosonic system may serve as hardware-efficient quantum resources for quantum computation and memory.   

Live

Bosonic codes[1] introduce a notion of a qubit in the context of infinite-dimensional Hilbert spaces. Namely, the Gottesman-Kitaev-Preskill (GKP) bosonic code[2] has several desirable properties[1] and has seen a sharp increase in interest over the past few years.

A problem with bosonic subspace encodings is that the code subspace is vanishingly small compared to the full CV Hilbert space. Another problem is that, for the GKP code, the codewords are unphysical. Further, their physical versions can have little overlap, yet represent the same logical information.

I introduce a formalism where a mode is divided into two virtual subsystems: a qubit and a mode. I show how, from this perspective, every continuous-variable state encodes qubits. I then give several demonstrations of how, with the subsystem approach, the tools from standard quantum computing are readily available to CV schemes. Finally, I apply the modular subsystem formalism to a real-world scheme for CV quantum computation-CV cluster states.

References
[1] Albert, Victor V., et al. - Physical Review A 97.3 (2018): 032346.
[2] Gottesman, Daniel, Alexei Kitaev, and John Preskill. - Physical Review A 64.1 (2001): 012310.   

Live

We propose a novel architecture of quantum-error-correction-based quantum repeaters that combines the techniques used in discrete and continuous variable quantum information. Specifically, we propose to encode the transmitted qubits in a concatenated code consisting of two levels. On the first level we use a continuous variable GKP code which encodes the qubit in a single bosonic mode. On the second level we use a small discrete variable code. Such an architecture introduces two major novelties which allow us to make efficient use of resources. Firstly, our architecture makes use of two types of quantum repeaters: the simpler GKP repeaters that need to only be able to store and correct errors on a single GKP qubit and more powerful but more costly multi-qubit repeaters that additionally can correct errors on the higher level. The combination of using the two types of repeaters enables us to achieve performance needed in practical scenarios with a significantly reduced cost with respect to an architecture based solely on multiqubit repeaters. Secondly, the use of the GKP code on the lower level provides us with the information about the success probability of the specific GKP correction round. This analog information offers significant boost in performance of our scheme.   

Live

It has been shown that universal control of an oscillator, realized here as a mode of a microwave cavity, can be achieved by coupling, in the strong-dispersive regime, the oscillator to an ancilla two-level system, such as a transmon [Heeres et al., 2017]. However, the rate of control appears to be limited by the interaction strength. In this two-part talk, we explore how displacements of the oscillator which are large on the scale of zero point fluctuations can generate an effective conditional displacement interaction [Campagne-Ibarcq et al., 2020], leading in turn to universal oscillator control with a speed limited by the oscillator drive strength rather than the strength of the oscillator-transmon coupling.

In this talk, we show that because displacements do not change the extent of states in phase space, they do not lead to additional decoherence. Furthermore, optimizing pulses based on conditional displacements is tractable because the simulations can be carried out in a displaced frame. We explore whether significantly faster pulses than the standard speed limit can be found with usual coupling strengths, thereby avoiding decoherence during logical gates.   

Live

It has been shown that universal control of an oscillator, realized here as a mode of a microwave cavity, can be achieved by coupling, in the strong-dispersive regime, the oscillator to an ancilla two-level system, such as a transmon [Heeres et al., 2017]. However, the rate of control appears to be limited by the interaction strength. In this two-part talk, we explore how displacements of the oscillator which are large on the scale of zero point fluctuations can generate an effective conditional displacement interaction [Campagne-Ibarcq et al., 2020], leading in turn to universal oscillator control with a speed limited by the oscillator drive strength rather than the strength of the oscillator-transmon coupling.

In this talk, we demonstrate how, even in the weak-dispersive regime, the conditional displacement gate can remain the generating gate of universal control. The advantage of working in this regime is that it minimizes unwanted nonlinearities and losses of the cavity due to the reverse Purcell effect. We also present practical pulse sequences that are robust against low-frequency transmon dephasing, as well as low-frequency amplitude drifts of the cavity drive.   

Live

Continuous-variable (CV) systems, most notably quantum harmonic oscillators, are ubiquitous in nature. CV quantum computing seeks to leverage their infinite-dimensional Hilbert space to perform quantum tasks. However, protection from noise native to the physical systems requires digitizing quantum information so that error correction can be performed. An encoding of digital quantum information into a continuous-variable system is called a bosonic code. Design and realization of useful bosonic codes, as well as their integration into fault-tolerant platforms, are spirited, active areas of research.

I will introduce the menagerie of bosonic codes, including a promising new class based on number-phase minimum uncertainty states, and highlight several cutting-edge experimental demonstrations. Then, I will describe how certain codes can be integrated into highly scalable optical platforms as a promising path towards a fault-tolerant quantum computer.   

Live

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. We show that a realistic three-wave mixing microwave architecture based on the superconducting nonlinear asymmetric inductive element [1] 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.
Ref. [1] Frattini et al., Appl. Phys. Lett. 110, 222603 (2017).   

Live

The Gottesman-Kitaev-Preskill (GKP) code is one of the most promising bosonic error correcting codes, and has recently been realized both in trapped ion and superconducting circuit experiments. An attractive feature of the GKP code is the fact that all Clifford gates can be implemented with quadratic Hamiltonians, while Pauli measurements can be performed using homodyne detection. We here moreover show that all single-qubit Clifford gates can be performed by simply updating the phase of local oscillators, thus reducing both software and hardware demands. To this end, we present a simple superconducting two-mode coupler that allows all Clifford gates to be executed using a single piece of hardware. We numerically estimate the gate fidelity for both ideal and realistic gate implementations.   

Live

Quantum computation requires coherent highly controllable gates with low error rates. A long-standing target is an exponential-swap gate which requires an ancilla qubit to entangle two microwave fields. A transmon ancilla has been previously used (Gao et al. Nature 566, 509 (2019)); it introduces a back propagation of ancilla errors into the fields. We propose a new way of doing a controlled beam splitter with a SNAIL-based Kerr cat that is transparent to the dominant error channel. We show that one can implement a beam splitter with a phase that depends on the state of the ancilla cat using suitable driving fields. Encoding quantum information into coherent states of the Kerr cat biases its decoherence towards phase flips, allowing significantly improved performance compared to the transmon setup. With this controlled beam splitter, it is possible to implement controlled-swap and further exponential-swap gates for microwave fields, which allows further applications such as swap tests or quantum random access memory protocols.   

Live

Large continuous-variable (CV) cluster states containing 1000s of modes have recently been used to experimentally demonstrate measurement-based single- and two-mode gate operations. A key primitive component of CV cluster states is a chain of two-mode squeezed states (TMSSs) called a macronode wire. Non-local, two-mode homodyne measurements are used to teleport an input state along the wire. Adjusting the measurement bases applies Gaussian gates to the input states via gate teleportation.

We extend this protocol by replacing the TMSSs in the macronode wire with more general two-mode entangled states. Teleporting through these states realizes non-Gaussian and potentially non-unitary CV gates—a powerful supplement to standard gate teleportation. We apply our general result to show how Gottesman-Kitaev-Preskill (GKP) error correction can be performed in a teleportation-based fashion in the macronode wire (and thus other CV cluster-state architectures) even when the GKP-state source is probabilistic.   

Live

Conical intersections (CIs) are ubiquitous features in quantum chemistry where two molecular potential energy surfaces intersect. Characterized by a breakdown of the Born-Oppenheimer approximation, they result in strong hybridization between electronic and nuclear degrees of freedom. CIs enable nonadiabatic transitions between electronic states and, when combined with vibrational damping, may heavily influence outcomes of chemical reactions. Though conventionally investigated through spectroscopic means, it would be desirable to engineer a synthetic CI with tunable control to systematically explore the parameter space. Superconducting circuits have emerged as a powerful platform for simulating quantum systems, possessing a versatile range of controllable interactions and engineered dissipation. In this talk, we report on experimental progress towards simulating a CI Hamiltonian in a circuit with one nonlinear (electronic) and two linear (nuclear) modes. We engineer the system to support two distinct ground states in one of the nuclear coordinates, representing reactant and product configurations. By preparing an excited state and tuning the dissipation, we highlight the competition between coherent evolution and damping in determining the branching ratio of the model reaction.   

Not Participating

Electromagnetic fields possess zero point fluctuations (ZPF) which lead to observable
effects such as the Lamb shift and the Casimir effect. In the traditional quantum optics domain, these corrections remain perturbative due to the smallness of the fine structure constant. To provide a direct observation of non-perturbative effects driven by ZPF in an open quantum system we wire a highly non-linear Josephson junction to a multi mode high impedance cavity (~4000 modes), allowing large phase fluctuations across the junction. Consequently, the resonance of the former acquires a relative frequency shift that is orders of magnitude larger than for natural atoms. Detailed modelling confirms that this renormalization is highly non-linear and quantum. Remarkably, the junction transfers its non-linearity to about 30 environmental modes, a striking back-action effect goes beyond the Kerr approximation. This work opens many exciting prospects for longstanding quests such as the tailoring of many-body Hamiltonians in the strongly non-linear regime, the observation of Bloch oscillations, highly efficient photon conversion or the development of high-impedance qubits.   

On Demand

Quantum walks (QWs) represent a powerful resource in different fields of physics due to its potential for simulating complex topological systems, paving the way to universal quantum computation. Moreover, two-dimensional QWs prove to outperform in the realization of quantum research algorithms and in the simulation of topological characteristics with respect to the one-dimensional counterpart. We present an innovative platform feasible for the realization of two-dimensional multiphoton QW in the transverse momentum of light. The key element of the platform is a new device called G-plate. When the photons propagate through these devices, they acquire a complex internal structure in the transverse momentum space. We experimentally realize the three-step dynamic of two walkers in different initial positions on the lattice within the quantum regime. Our platform allows us to control the degree of indistinguishability of photons, allowing us to investigate the bosonic behavior of two photons in the QW dynamic. As a future prospect, our platform could be exploited for the implementation of more complex schemes using multiphoton interference, such as multiphoton Boson sampling and the hybrid entanglement state engineering.