Bosonic encoding is an approach for quantum information processing, promising lower hardware overhead by encoding in the many levels of a harmonic-oscillator mode. Scaling to multiple modes requires weak interaction for independent control, yet strong interaction for fast control. Applying fast and efficient universal control on multiple modes remains an open problem. Surprisingly, we find that displacements conditioned on the state of a single-qubit ancilla coupled to multiple harmonic oscillators are sufficient for universal control. We present the conditional- operation concept, which can be used to reduce the duration of entangling gates. Within this guiding concept, we develop the conditional- displacement control method which enables fast generation and control of bosonic states in multimode systems weakly coupled to a single-ancilla qubit. Our method is fast despite the weak ancilla coupling. The weak coupling in turn allows for excellent separability and thus independent control. We demonstrate our control on a superconducting transmon qubit weakly coupled to a multimode superconducting cavity. We create both entangled and separable cat states in different modes of the multimode cavity, showing entangling operations at low crosstalk while maintaining independent control of the different modes. We show that the operation time is not limited by the inverse of the coupling rate, which is the typical timescale, and we exceed it by almost 2 orders of magnitude. We verify our results with an efficient method for measurement of the multimode characteristic function which employs our conditional- displacement. Our results inspire a new approach toward general entangling operations and allow for fast and efficient multimode bosonic encoding and measurement. Published by the American Physical Society 2024
Bosonic encoding is an approach for quantum information processing, promising lower hardware overhead by encoding in the many levels of a harmonic-oscillator mode. Scaling to multiple modes requires weak interaction for independent control, yet strong interaction for fast control. Applying fast and efficient universal control on multiple modes remains an open problem. Surprisingly, we find that displacements conditioned on the state of a single-qubit ancilla coupled to multiple harmonic oscillators are sufficient for universal control. We present the conditional- operation concept, which can be used to reduce the duration of entangling gates. Within this guiding concept, we develop the conditional- displacement control method which enables fast generation and control of bosonic states in multimode systems weakly coupled to a single-ancilla qubit. Our method is fast despite the weak ancilla coupling. The weak coupling in turn allows for excellent separability and thus independent control. We demonstrate our control on a superconducting transmon qubit weakly coupled to a multimode superconducting cavity. We create both entangled and separable cat states in different modes of the multimode cavity, showing entangling operations at low crosstalk while maintaining independent control of the different modes. We show that the operation time is not limited by the inverse of the coupling rate, which is the typical timescale, and we exceed it by almost 2 orders of magnitude. We verify our results with an efficient method for measurement of the multimode characteristic function which employs our conditional- displacement. Our results inspire a new approach toward general entangling operations and allow for fast and efficient multimode bosonic encoding and measurement. Published by the American Physical Society 2024
High-coherence superconducting cavities offer a hardware-efficient platform for quantum information processing. To achieve universal operations of these bosonic modes, the requisite nonlinearity is realized by coupling them to a transmon ancilla. However, this configuration is susceptible to crosstalk errors in the dispersive regime, where the ancilla frequency is Stark shifted by the state of each coupled bosonic mode. This leads to a frequency mismatch of the ancilla drive, lowering the gate fidelities. To mitigate such coherent errors, we employ quantum optimal control to engineer ancilla pulses that are robust to the frequency shifts. These optimized pulses are subsequently integrated into a recently developed echoed conditional displacement protocol for executing single- and two-mode operations. Through numerical simulations, we examine two representative scenarios: the preparation of single-mode Fock states in the presence of spectator modes and the generation of two-mode entangled Bell-cat states. Our approach markedly suppresses crosstalk errors, outperforming conventional ancilla control methods by orders of magnitude. These results provide guidance for experimentally achieving high-fidelity multimode operations and pave the way for developing high-performance bosonic quantum information processors.
Bosonic quantum error correction represents a route towards significant hardware overhead reduction for fault-tolerant quantum computation. However, controlling quantum harmonic oscillators that encode the quantum information comes with its own challenges. In many platforms such as superconducting circuits and trapped ions, the bosonic degree of freedom is controlled by an ancillary qubit whose decay during a qubit-oscillator gate propagates to the oscillator as a continuous error. Such errors generally lie beyond the scope of errors that the bosonic code is capable of correcting. We introduce the continuous-variable flag: an auxiliary oscillator coupled to the ancilla whose phase-space position encodes the continuous error corrupting the storage mode. After the gate, a heterodyne measurement of the flag estimates this continuous-variable error and a subsequent feedback control undoes it up to a bosonic code stabilizer. The scheme requires no new ingredients, as the flag is driven by the same dispersive coupling, displacements and rotations that implement the gates it is supposed to protect. We construct sequences protecting conditional rotations to all orders in ancilla decay with one flag, and conditional displacements to first order with one flag and to all orders with two. Simulations with realistic circuit-QED parameters predict more than three orders of magnitude fewer logical errors for the parity measurement of a four-component cat code, and a finite-energy GKP logical lifetime near its noiseless-ancilla bound. By discretizing continuous-variable errors, the flags introduced in this work are a useful tool for any hybrid oscillator-qubit processor, and open the route towards fault-tolerant simulation and computation with bosonic modes.
Encoding a qubit in a high quality superconducting microwave cavity offers the opportunity to perform the first layer of error correction in a single device, but presents a challenge: how can quantum oscillators be controlled while introducing a minimal number of additional error channels? We focus on the two-qubit portion of this control problem by using a 3-wave mixing coupling element to engineer a programmable beamsplitter interaction between two bosonic modes separated by more than an octave in frequency, without introducing major additional sources of decoherence. Combining this with single-oscillator control provided by a dispersively coupled transmon provides a framework for quantum control of multiple encoded qubits. The beamsplitter interaction gbs is fast relative to the timescale of oscillator decoherence, enabling over 103 beamsplitter operations per coherence time, and approaching the typical rate of the dispersive coupling χ used for individual oscillator control. Further, the programmable coupling is engineered without adding unwanted interactions between the oscillators, as evidenced by the high on-off ratio of the operations, which can exceed 105. We then introduce a new protocol to realize a hybrid controlled-SWAP operation in the regime gbs≈χ, in which a transmon provides the control bit for the SWAP of two bosonic modes. Finally, we use this gate in a SWAP test to project a pair of bosonic qubits into a Bell state with measurement-corrected fidelity of 95.5%±0.2%.
Fault-tolerant quantum computation with bosonic qubits often necessitates the use of noisy discrete-variable ancillae. In this work, we establish a comprehensive and practical fault-tolerance framework for such a hybrid system and synthesize it with fault-tolerant protocols by combining bosonic quantum error correction (QEC) and advanced quantum control techniques. We introduce essential building blocks of error-corrected gadgets by leveraging ancilla-assisted bosonic operations using a generalized variant of path-independent quantum control. Using these building blocks, we construct a universal set of error-corrected gadgets that tolerate a single-photon loss and an arbitrary ancilla fault for four-legged cat qubits. Notably, our construction requires only dispersive coupling between bosonic modes and ancillae, as well as beam-splitter coupling between bosonic modes, both of which have been experimentally demonstrated with strong strengths and high accuracy. Moreover, each error-corrected bosonic qubit is comprised of only a single bosonic mode and a three-level ancilla, featuring the hardware efficiency of bosonic QEC in the full fault-tolerant setting. We numerically demonstrate the feasibility of our schemes using current experimental parameters in the circuit-QED platform. Finally, we present a hardware-efficient architecture for fault-tolerant quantum computing by concatenating the four-legged cat qubits with an outer qubit code utilizing only beam-splitter couplings. Our estimates suggest that the overall noise threshold can be reached using existing hardware. These developed fault-tolerant schemes extend beyond their applicability to four-legged cat qubits and can be adapted for other rotation-symmetrical codes, offering a promising avenue toward scalable and robust quantum computation with bosonic qubits. Published by the American Physical Society 2024
The Jaynes-Cummings (JC) interaction-the coherent exchange of excitations between a two-level system and a harmonic oscillator-is one of the fundamental interactions of quantum optics, realized across platforms such as cavity quantum electrodynamics, trapped ions, mechanical resonators, and superconducting circuits. Although JC interactions and qubit rotations form a universal gate set for oscillator control, practical implementations have not been demonstrated. Here we develop and experimentally demonstrate universal JC-based oscillator control by compiling arbitrary unitary gates into sequences of JC interactions and qubit rotations. In our experiment, the oscillator is realized using a mode of a high quality factor microwave cavity and the ancilla qubit using a superconducting transmon circuit, with the JC interaction implemented by a sideband interaction enabled by the Josephson nonlinearity. The native gates are constructed to be closed below a chosen cutoff photon number, encoding a qudit with suppressed leakage errors, while ancilla relaxation errors are detectable. We further find that the dispersive shift serves as a compilation resource that reduces circuit depths. We demonstrate universal qudit control and implement a single-qutrit gate set with a mean post-selected process fidelity of 96.3%, as well as ququart and ququint shift gates. These results establish Jaynes-Cummings control as a practical route to universal oscillator control, enabling programmable bosonic processors across a variety of quantum platforms.
Quantum computers use the principles of quantum mechanics to store and manipulate information encoded in the quantized levels of physical systems. A goal of quantum computers is to accurately solve problems that are intractable for classical computers, such as simulating molecular dynamics and prime factorization of large integers. However, the failure probability is too high for quantum computers to solve useful problems. Fortunately, quantum error correction (QEC) provides a way to perform robust computations in the presence of errors by distributing information across multiple physical systems. Bosonic quantum error-correcting codes offer a hardware-efficient avenue for QEC where information is encoded in the continuous phase-space of a single harmonic oscillator. The linear harmonic oscillator realised as the microwave modes of a superconducting resonator is controlled via a nonlinear superconducting control qubit. Experiments on bosonic QEC are limited by bit flip errors in the control qubit, leading to low logical lifetimes of the storage resonator. Our idea is to replace the transmon with a heavy fluxonium qubit that has intrinsic protection against bit-flips. With this architecture in mind, we investigate the performance of a universal gateset comprising of Echoed Conditional Displacement gates and single qubit rotations. Wenumerically solve the Schrödinger equation for a closed resonator- fluxonium device and use a combination of quantum and semiclassical approaches to simulate the system exactly. We adopt a pulse-level implementation of quantum gates where we realize target gates by applying tailored drives on the resonator. We develop a resonator frame-tracking technique that allows us to simulate the dynamics of the driven resonator exactly with less than five fock levels while retaining all the physics of the problem. This technique allowed us to accurately simulate the device up to a few 10000s of photons, matching experimental results, that otherwise would not be possible even using high-performance computers due to excessive memory and simulation time required.