A stabilized logical quantum bit encoded in grid states of a superconducting cavity (original) (raw)

Quantum codes for simplifying design and suppressing decoherence in superconducting phase-qubits

2002

We introduce simple qubit-encodings and logic gates which eliminate the need for certain difficult single-qubit operations in superconducting phase-qubits, while preserving universality. The simplest encoding uses two physical qubits per logical qubit. Two architectures for its implementation are proposed: one employing N physical qubits out of which N/2 are ancillas fixed in the |1 state, the other employing N/2 + 1 physical qubits, one of which is a bus qubit connected to all others. Details of a minimal set of universal encoded logic operations are given, together with recoupling schemes, that require nanosecond pulses. A generalization to codes with higher ratio of number of logical qubits per physical qubits is presented. Compatible decoherence and noise suppression strategies are also discussed.

Extending the lifetime of a quantum bit with error correction in superconducting circuits

Nature, 2016

Quantum error correction (QEC) can overcome the errors experienced by qubits and is therefore an essential component of a future quantum computer. To implement QEC, a qubit is redundantly encoded in a higher-dimensional space using quantum states with carefully tailored symmetry properties. Projective measurements of these parity-type observables provide error syndrome information, with which errors can be corrected via simple operations. The 'break-even' point of QEC-at which the lifetime of a qubit exceeds the lifetime of the constituents of the system-has so far remained out of reach. Although previous works have demonstrated elements of QEC, they primarily illustrate the signatures or scaling properties of QEC codes rather than test the capacity of the system to preserve a qubit over time. Here we demonstrate a QEC system that reaches the break-even point by suppressing the natural errors due to energy loss for a qubit logically encoded in superpositions of Schrödinger-c...

Protecting a bosonic qubit with autonomous quantum error correction

Nature, 2021

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 operations 2-7 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 system 1,8-14 , but so far it has remained challenging to achieve the specific form of dissipation to counter the most prominent errors in a physical platform. Here we encode a logical qubit in Schrödinger cat-like multiphoton states 15 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 realizes autonomous correction against single-photon loss and boosts the coherence time of the multiphoton 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 techniques 16-18 , our experiment suggests reservoir engineering as a resource-efficient alternative or supplement to active QEC in future quantum computing architectures.

Preparation and manipulation of a fault-tolerant superconducting qubit

Physical Review B, 2007

We describe a qubit encoded in continuous quantum variables of an rf superconducting quantum interference device. Since the number of accessible states in the system is infinite, we may protect its two-dimensional subspace from small errors introduced by the interaction with the environment and during manipulations. We show how to prepare the fault-tolerant state and manipulate the system. The discussed operations suffice to perform quantum computation on the encoded state, syndrome extraction, and quantum error correction. We also comment on the physical sources of errors and possible imperfections while manipulating the system.

Realization of three-qubit quantum error correction with superconducting circuits

Quantum computers could be used to solve certain problems exponentially faster than classical computers, but are challenging to build because of their increased susceptibility to errors. However, it is possible to detect and correct errors without destroying coherence, by using quantum error correcting codes 1. The simplest of these are three-quantum-bit (three-qubit) codes, which map a one-qubit state to an entangled three-qubit state; they can correct any single phase-flip or bit-flip error on one of the three qubits, depending on the code used 2. Here we demonstrate such phase-and bit-flip error correcting codes in a superconducting circuit. We encode a quantum state 3,4 , induce errors on the qubits and decode the error syndrome—a quantum state indicating which error has occurred—by reversing the encoding process. This syndrome is then used as the input to a three-qubit gate that corrects the primary qubit if it was flipped. As the code can recover from a single error on any qubit, the fidelity of this process should decrease only quadratically with error probability. We implement the correcting three-qubit gate (known as a conditional-conditional NOT, or Toffoli, gate) in 63 nanoseconds, using an interaction with the third excited state of a single qubit. We find 85 6 1 per cent fidelity to the expected classical action of this gate, and 78 6 1 per cent fidelity to the ideal quantum process matrix. Using this gate, we perform a single pass of both quantum bit-and phase-flip error correction and demonstrate the predicted first-order insensitivity to errors. Concatenation of these two codes in a nine-qubit device would correct arbitrary single-qubit errors. In combination with recent advances in superconducting qubit coherence times 5,6 , this could lead to scalable quantum technology. Quantum error correction relies on detecting the presence of errors without gaining knowledge of the encoded quantum state. In the three-qubit error-correcting code, the subspace of the two additional 'ancilla' qubits uniquely encodes which of the four possible single-qubit errors has occurred, including the possibility of no flip. Crucially, errors consisting of finite rotations can also be corrected using these schemes because the error syndromes are allowed to be in superpositions of the possible outcomes, flipped and not flipped 2. Previous works implementing error correcting codes in liquid-7–9 and solid-state 10 NMR and with trapped ions 11,12 have demonstrated two possible strategies for using the error syndromes. The first is to measure the ancillas (thereby projecting the syndrome) and use a classical logic operation to correct the detected error. This 'feed-forward' capability is challenging in superconducting circuits as it requires a fast and high-fidelity quantum non-demolition measurement, but is probably a necessary component to achieve scalable fault tolerance 2,13. The second strategy, as recently demonstrated with trapped ions 12 and used here, is to replace the classical logic with a quantum controlled-controlled NOT (CCNOT) gate that performs the correction coherently, leaving the entropy associated with the error in the ancilla qubits, which can then be reset 14 if the code is to be repeated. The CCNOT gate performs exactly the action that would follow the measurement in the first scheme: flipping the primary qubit if and only if the ancillas encode the associated error syndrome. The CCNOT gate is also vital for a wide variety of applications such as Shor's factoring algorithm 15 and has attracted substantial experimental interest, with recent implementations in linear optics 16 , trapped ions 17 and superconducting circuits 18,19. Here we use the circuit quantum electrodynamics architecture 20 to couple four super-conducting transmon qubits 21 to a single microwave cavity bus 22 , where each qubit transition frequency can be controlled on nano-second timescales with individual flux bias lines 23 and collectively measured by interrogating transmission through the cavity 24. (The details of the device can be found in Methods Summary and in ref. 3.) The frequencies of the qubits, labelled Q 1 –Q 4 , are tuned respectively to 6, 7, 7.85 and ,13 GHz, with Q 4 unused. In this Letter, we first demonstrate the three-qubit interaction used in the gate, which is an extension of interactions used in previous two-qubit gates 3,23,25 , and show how this interaction can be used to create the desired CCNOT gate. We then verify its action and use it to demonstrate error correction for an error on a single qubit with the bit-flip code and then for simultaneous errors on all three qubits with the phase-flip code. We find a quadratic dependence of process fidelity on error probability, indicating that the algorithm is correcting errors as predicted. Our three-qubit gate uses an interaction with the third excited state of one transmon. Specifically, it relies on the unique capability among computational states (eigenstates of the Pauli operator Z) of j111ae to interact with the non-computational state j003ae (the notation jabcae refers to the excitation levels of Q 1 –Q 3 , respectively). As the direct interaction of these states is prohibited to first order, we first transfer the quantum amplitude of j111ae to the intermediate state j102ae, which itself couples strongly to j003ae. Calculated energy levels and time-domain data showing interaction between j011ae and j002ae (which is identical to that between j111ae and j102ae except for a 6-GHz offset) as a function of the flux bias on Q 2 are shown in Fig. 1a. Once the amplitude of j111ae has been transferred to j102ae with a sudden swap interaction , a three-qubit phase is acquired by moving Q 1 up in frequency adiabatically, near the avoided crossing with j003ae. Figure 1b shows the avoided crossing between these states as a function of the flux bias on Q 1. This crossing shifts the frequency of j102ae relative to the sum of the frequencies of j100ae and j002ae to yield the three-qubit phase. The detailed procedure of the gate is shown in Fig. 2a, and is implemented in 63 ns. Further details can be found in Supplementary Information. We demonstrate the gate by first measuring its classical action. The controlled-controlled phase (CCPhase) gate, which maps j111ae to 2j111ae, has no effect on pure computational states so we implement a CCNOT gate by concatenating pre-and post-gate rotations on Q 2 , as indicated in the unshaded regions of Fig. 2a. Such a gate ideally swaps j101ae and j111ae and does nothing to the remaining states. To verify this, we prepare the eight computational states, implement the gate and measure its output using three-qubit state tomography 3 to generate the classical truth table. The intended state is reached with 85 6 1% fidelity on average. This measurement is sensitive only to classical action, however, so we complete our verification by performing full quantum process tomography on the CCPhase gate, which can detect

Stabilizing Rabi oscillations in a superconducting qubit using quantum feedback

Nature, 2012

The act of measurement bridges the quantum and classical worlds by projecting a superposition of possible states into a single, albeit probabilistic, outcome. The time-scale of this "instantaneous" process can be stretched using weak measurements 1,2 so that it takes the form of a gradual random walk towards a final state. Remarkably, the interim measurement record is sufficient to continuously track and steer the quantum state using feedback 3-8 . Here, we report the first implementation of quantum feedback control in a solid state system, in our case a superconducting quantum bit (qubit) coupled to a microwave cavity 9 . Probing the state of the cavity with less than one photon on average, implements a weak measurement of the qubit state. These photons are then directed to a high-bandwidth quantum-noise-limited amplifier 10,11 , which enables realtime monitoring of the state of the cavity-and hence that of the qubit-with high fidelity. We demonstrate quantum feedback control by inhibiting the decay of Rabi oscillations, allowing them to persist indefinitely 12 . This new ability permits active suppression of decoherence and defines a path for continuous quantum error correction 13,14 . Other novel avenues include quantum state stabilization 4,7,15 , entanglement generation using measurement 16 , state purification 17 , and adaptive measurements 18,19 . arXiv:1205.5591v1 [cond-mat.mes-hall]

New class of quantum error-correcting codes for a bosonic mode

We construct a new class of quantum error-correcting codes for a bosonic mode which are advantageous for applications in quantum memories, communication, and scalable computation. These 'binomial quantum codes' are formed from a finite superposition of Fock states weighted with binomial coefficients. The binomial codes can exactly correct errors that are polynomial up to a specific degree in bosonic creation and annihilation operators, including amplitude damping and displacement noise as well as boson addition and dephasing errors. For realistic continuous-time dissipative evolution, the codes can perform approximate quantum error correction to any given order in the timestep between error detection measurements. We present an explicit approximate quantum error recovery operation based on projective measurements and unitary operations. The binomial codes are tailored for detecting boson loss and gain errors by means of measurements of the generalized number parity. We discuss optimization of the binomial codes and demonstrate that by relaxing the parity structure, codes with even lower unrecoverable error rates can be achieved. The binomial codes are related to existing two-mode bosonic codes but offer the advantage of requiring only a single bosonic mode to correct amplitude damping as well as the ability to correct other errors. Our codes are similar in spirit to 'cat codes' based on superpositions of the coherent states, but offer several advantages such as smaller mean number, exact rather than approximate orthonormality of the code words, and an explicit unitary operation for repumping energy into the bosonic mode. The binomial quantum codes are realizable with current superconducting circuit technology and they should prove useful in other quantum technologies, including bosonic quantum memories, photonic quantum communication, and optical-to-microwave up-and down-conversion.

Gated Conditional Displacement Readout of Superconducting Qubits

Physical Review Letters, 2019

We have realized a new interaction between superconducting qubits and a readout cavity that results in the displacement of a coherent state in the cavity, conditioned on the state of the qubit. This conditional state, when it reaches the cavity-following, phase-sensitive amplifier, matches its measured observable, namely the in-phase quadrature. In a setup where several qubits are coupled to the same readout resonator, we show it is possible to measure the state of a target qubit with minimal dephasing of the other qubits. Our results suggest novel directions for faster readout of superconducting qubits and implementations of bosonic quantum error-correcting codes.