Multimode cat codes

  1. Victor V. Albert,
  2. Shantanu O. Mundhada,
  3. Alexander Grimm,
  4. Steven Touzard,
  5. Michel H. Devoret,
  6. and Liang Jiang
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 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 or gains in any one mode.

Quantum Channel Construction with Circuit Quantum Electrodynamics

  1. Chao Shen,
  2. Kyungjoo Noh,
  3. Victor V. Albert,
  4. Stefan Krastanov,
  5. Michel H. Devoret,
  6. Robert J. Schoelkopf,
  7. S. M. Girvin,
  8. and Liang Jiang
Quantum channels can describe all transformations allowed by quantum mechanics. We provide an explicit universal protocol to construct all possible quantum channels, using a single
qubit ancilla with quantum non-demolition readout and adaptive control. Our construction is efficient in both physical resources and circuit depth, and can be demonstrated using superconducting circuits and various other physical platforms. There are many applications of quantum channel construction, including system stabilization and quantum error correction, Markovian and exotic channel simulation, implementation of generalized quantum measurements and more general quantum instruments. Efficient construction of arbitrary quantum channels opens up exciting new possibilities for quantum control, quantum sensing and information processing tasks.

Cat codes with optimal decoherence suppression for a lossy bosonic channel

  1. Linshu Li,
  2. Chang-ling Zou,
  3. Victor V. Albert,
  4. Sreraman Muralidharan,
  5. S. M. Girvin,
  6. and Liang Jiang
We investigate cat codes that can correct multiple excitation losses and identify two types of logical errors: bit-flip errors due to excessive excitation loss and dephasing errors
due to quantum back-action from the environment. We show that selected choices of logical subspace and coherent amplitude can efficiently reduce dephasing errors. The trade-off between the two major errors enables optimized performance of cat codes in terms of minimized decoherence. With high coupling efficiency, we show that one-way quantum repeaters with cat codes feature drastically boosted secure communication rate per mode compared with conventional encoding schemes, and thus showcase the promising potential of quantum information processing with continuous variable quantum codes.

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

  1. Marios H. Michael,
  2. Matti Silveri,
  3. R. T. Brierley,
  4. Victor V. Albert,
  5. Juha Salmilehto,
  6. Liang Jiang,
  7. and S. M. Girvin
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.

Cavity State Manipulation Using Photon-Number Selective Phase Gates

  1. Reinier W. Heeres,
  2. Brian Vlastakis,
  3. Eric Holland,
  4. Stefan Krastanov,
  5. Victor V. Albert,
  6. Luigi Frunzio,
  7. Liang Jiang,
  8. and Robert J. Schoelkopf
The large available Hilbert space and high coherence of cavity resonators makes these systems an interesting resource for storing encoded quantum bits. To perform a quantum gate on
this encoded information, however, complex nonlinear operations must be applied to the many levels of the oscillator simultaneously. In this work, we introduce the Selective Number-dependent Arbitrary Phase (SNAP) gate, which imparts a different phase to each Fock state component using an off-resonantly coupled qubit. We show that the SNAP gate allows control over the quantum phases by correcting the unwanted phase evolution due to the Kerr effect. Furthermore, by combining the SNAP gate with oscillator displacements, we create a one-photon Fock state with high fidelity. Using just these two controls, one can construct arbitrary unitary operations, offering a scalable route to performing logical manipulations on oscillator-encoded qubits.

Universal Control of an Oscillator with Dispersive Coupling to a Qubit

  1. Stefan Krastanov,
  2. Victor V. Albert,
  3. Chao Shen,
  4. Chang-Ling Zou,
  5. Reinier W. Heeres,
  6. Brian Vlastakis,
  7. Robert J. Schoelkopf,
  8. and Liang Jiang
We investigate quantum control of an oscillator mode off-resonantly coupled to an ancillary qubit. In the strong dispersive regime, we may drive the qubit conditioned on number states
of the oscillator, which together with displacement operations can achieve universal control of the oscillator. Based on our proof of universal control, we provide explicit constructions for arbitrary state preparation and arbitrary unitary operation of the oscillator. Moreover, we present an efficient procedure to prepare the number state ∣∣n⟩ using only O(n‾‾√) operations. We also compare our scheme with known quantum control protocols for coupled qubit-oscillator systems. This universal control scheme of the oscillator can readily be implemented using superconducting circuits.

Dynamically protected cat-qubits: a new paradigm for universal quantum computation

  1. Mazyar Mirrahimi,
  2. Zaki Leghtas,
  3. Victor V. Albert,
  4. Steven Touzard,
  5. Robert J. Schoelkopf,
  6. Liang Jiang,
  7. and Michel H. Devoret
We present a new hardware-efficient paradigm for universal quantum computation which is based on encoding, protecting and manipulating quantum information in a quantum harmonic oscillator.
This proposal exploits multi-photon driven dissipative processes to encode quantum information in logical bases composed of Schr\“odinger cat states. More precisely, we consider two schemes. In a first scheme, a two-photon driven dissipative process is used to stabilize a logical qubit basis of two-component Schr\“odinger cat states. While such a scheme ensures a protection of the logical qubit against the photon dephasing errors, the prominent error channel of single-photon loss induces bit-flip type errors that cannot be corrected. Therefore, we consider a second scheme based on a four-photon driven dissipative process which leads to the choice of four-component Schr\“odinger cat states as the logical qubit. Such a logical qubit can be protected against single-photon loss by continuous photon number parity measurements. Next, applying some specific Hamiltonians, we provide a set of universal quantum gates on the encoded qubits of each of the two schemes. In particular, we illustrate how these operations can be rendered fault-tolerant with respect to various decoherence channels of participating quantum systems. Finally, we also propose experimental schemes based on quantum superconducting circuits and inspired by methods used in Josephson parametric amplification, which should allow to achieve these driven dissipative processes along with the Hamiltonians ensuring the universal operations in an efficient manner.