Abstract Bosonic quantum error correcting codes are primarily designed to protect against single-photon loss. To correct for this type of error, one can encode the logical qubit in code spaces with a definite photon parity, such as cat codes or binomial codes. Error correction requires a recovery operation that maps the error states—which have opposite parity—back onto the code space. Here, we realize a collection of photon-number-selective, simultaneous photon addition operations on a bosonic mode, a microwave cavity, assisted by a superconducting qubit. These operations are implemented as two-photon transitions that excite the cavity and the qubit at the same time. The additional degree of freedom of the qubit makes it possible to implement a coherent, unidirectional mapping between spaces of opposite photon parity. We present the successful experimental implementation of the drives and the phase control they enable on superpositions of Fock states. The presented technique, when supplemented with qubit reset, is suitable for autonomous quantum error correction in bosonic systems and, more generally, opens the possibility to realize a range of non-unitary transformations on a bosonic mode.
Abstract Bosonic quantum error correcting codes are primarily designed to protect against single-photon loss. To correct for this type of error, one can encode the logical qubit in code spaces with a definite photon parity, such as cat codes or binomial codes. Error correction requires a recovery operation that maps the error states—which have opposite parity—back onto the code space. Here, we realize a collection of photon-number-selective, simultaneous photon addition operations on a bosonic mode, a microwave cavity, assisted by a superconducting qubit. These operations are implemented as two-photon transitions that excite the cavity and the qubit at the same time. The additional degree of freedom of the qubit makes it possible to implement a coherent, unidirectional mapping between spaces of opposite photon parity. We present the successful experimental implementation of the drives and the phase control they enable on superpositions of Fock states. The presented technique, when supplemented with qubit reset, is suitable for autonomous quantum error correction in bosonic systems and, more generally, opens the possibility to realize a range of non-unitary transformations on a bosonic mode.
Abstract To build a universal quantum computer from fragile physical qubits, effective implementation of quantum error correction (QEC)1 is an essential requirement and a central challenge. Existing demonstrations of QEC are based on an active schedule of error-syndrome measurements and adaptive recovery operations2, 3, 4, 5, 6, 7 that are hardware intensive and prone to introducing and propagating errors. In principle, QEC can be realized autonomously and continuously by tailoring dissipation within the quantum system1, 8, 9, 10, 11, 12, 13, 14, but so far it has remained challenging to achieve the specific form of dissipation required to counter the most prominent errors in a physical platform. Here we encode a logical qubit in Schrödinger cat-like multiphoton states15 of a superconducting cavity, and demonstrate a corrective dissipation process that stabilizes an error-syndrome operator: the photon number parity. Implemented with continuous-wave control fields only, this passive protocol protects the quantum information by autonomously correcting single-photon-loss errors and boosts the coherence time of the bosonic qubit by over a factor of two. Notably, QEC is realized in a modest hardware setup with neither high-fidelity readout nor fast digital feedback, in contrast to the technological sophistication required for prior QEC demonstrations. Compatible with additional phase-stabilization and fault-tolerant techniques16, 17, 18, our experiment suggests quantum dissipation engineering as a resource-efficient alternative or supplement to active QEC in future quantum computing architectures.
The rich dynamics and large Hilbert space of quantum harmonic oscillators make them natural candidates for hardware-efficient and error-correctable quantum information processing. However, implementing direct entangling operations between oscillators remains an outstanding challenge. Existing strategies typically rely on parametrically activating interactions that populate the excited states of a nonlinear element, which introduces additional dissipation channels and potential leakage from the encoded manifold. Here, we engineer a Raman-assisted cross-Kerr interaction between microwave photons hosted in two superconducting cavities. Crucially, this dynamics does not excite the mediating nonlinear coupler, thereby suppressing coupler induced decoherence and leakage out of the bosonic code space. We use this direct nonlinear coupling to implement a controlled-phase gate within the single- and two-photon subspaces of two oscillators, deterministically generating entanglement between them. Finally, we use these engineered dynamics to implement a photon-number parity check on a storage cavity via purely bosonic interactions with an ancillary cavity, demonstrating an enhancement in the storage lifetime. Our work provides a promising pathway toward engineering robust operations that act entirely within a protected bosonic code space and realizing fault-tolerant quantum information processing with bosonic elements.
Abstract We introduce a driven-dissipative two-mode bosonic system whose reservoir causes simultaneous loss of two photons in each mode and whose steady states are superpositions of pair-coherent/Barut-Girardello coherent states. We show how quantum information encoded in a steady-state subspace of this system is exponentially immune to phase drifts (cavity dephasing) in both modes. Additionally, it is possible to protect information from arbitrary photon loss in either (but not simultaneously both) of the modes by continuously monitoring the difference between the expected photon numbers of the logical states. Despite employing more resources, the two-mode scheme enjoys two advantages over its one-mode cat-qubit counterpart with regards to implementation using current circuit QED technology. First, monitoring the photon number difference can be done without turning off the currently implementable dissipative stabilizing process. Second, a lower average photon number per mode is required to enjoy a level of protection at least as good as that of the cat-codes. We discuss circuit QED proposals to stabilize the code states, perform gates, and protect against photon loss via either active syndrome measurement or an autonomous procedure. We introduce quasiprobability distributions allowing us to represent two-mode states of fixed photon number difference in a two-dimensional complex plane, instead of the full four-dimensional two-mode phase space. The two-mode codes are generalized to multiple modes in an extension of the stabilizer formalism to non-diagonalizable stabilizers. The M -mode codes can protect against either arbitrary photon losses in up to M − 1 modes or arbitrary losses and gains in any one mode.
Bosonic codes offer a hardware-efficient strategy for quantum error correction by redundantly encoding quantum information in the large Hilbert space of a harmonic oscillator. However, experimental realizations of these codes are often limited by ancilla errors propagating to the encoded logical qubit during syndrome measurements. The Kerr-cat qubit has been proposed as an ancilla for these codes due to its theoretically-exponential noise bias, which would enable fault-tolerant error syndrome measurements, but the coupling required to perform these syndrome measurements has not yet been demonstrated. In this work, we experimentally realize driven parametric coupling of a Kerr-cat qubit to a high-quality-factor microwave cavity and demonstrate a gate set that would enable universal quantum control of the cavity. We measure the decoherence of the cavity in the presence of the Kerr-cat and discover excess dephasing due to heating of the Kerr-cat to excited states. By engineering frequency-selective dissipation to counteract this heating, we are able to eliminate this dephasing, thereby demonstrating a high on-off ratio of control. Our results pave the way toward using the Kerr-cat to fault-tolerantly measure error syndromes of bosonic codes. Biased noise qubits, which can selectively suppress certain types of noise, are advantageous for quantum error correction of bosonic codes. Here the authors make an important step in this direction by demonstrating quantum control of a harmonic oscillator with a biased noise qubit.
Abstract The rich dynamics and large Hilbert space of quantum harmonic oscillators make them natural candidates for hardware-efficient and error-correctable quantum information processing. However, implementing direct entangling operations between oscillators remains an outstanding challenge. Existing strategies typically rely on parametrically activating interactions that necessarily populate the excited states of a nonlinear coupler, introducing additional dissipation channels and potential leakage outside the encoded manifold. Here, we present a fundamentally different approach to generate nonlinear coupling between bosonic modes: a Raman-assisted cross-Kerr interaction between microwave photons hosted in two superconducting cavities, which does not populate the mediating coupler. Crucially, this dynamics preserves the bosonic code space throughout, which is essential for fault-tolerant bosonic quantum computation. We demonstrate that the engineered cross-Kerr can be used to directly implement controlled-phase gates between two oscillators encoded in the single- and two-photon subspaces. Further, we apply the engineered dynamics to implement code-space-preserving parity checks using one of the oscillators as an ancillary mode, demonstrating an all-bosonic syndrome extraction that extends the effective lifetime of a storage oscillator. This work establishes a qualitatively different strategy for bosonic control, providing a pathway toward fault-tolerant quantum information processing with purely bosonic gates and syndrome measurements.
Schr\{o}dinger cat states are crucial for exploration of fundamental issues
of quantum mechanics and have important applications in quantum information
processing. Here, we propose and experimentally demonstrate a method for
manipulating cat states in a cavity with the Aharonov-Anandan phase acquired by
a superconducting qubit, which is dispersively coupled to the cavity. Based on
this dispersive coupling, the qubit can be forced to trace out a circuit in the
projective Hilbert space conditional on one coherent state. By preparing the
cavity in a superposition of two coherent states, the geometric phase
associated with this transport is encoded to the relative probability amplitude
of these two coherent states. We demonstrate the photon-number parity of a cat
state in a cavity can be controlled by adjusting this geometric phase, which
offers the possibility for protecting its quantum coherence from single-photon
loss. Based on this geometric effect, we realize phase gates for one and two
photonic qubits whose logical basis states are encoded in two quasi-orthogonal
coherent states. We further demonstrate two-cavity gates with symmetric and
asymmetric Fock state encoding schemes. Our method can be directly extended to
implementation of controlled-phase gates between error-correctable logical
qubits.