The desired effective interaction for C D ^ gate is ∝ ( a ^ † + a ^ ) Z ^, where Z ^ is the Pauli operator of the transmon or Kerr-cat qubit. The frequencies of the resonator, transmon, and SNAIL are ω c, ω t, and ω s, respectively. Implementation of controlled-displacement gate, C D ^ ( ζ ), between a GKP resonator and a transmon (a) and Kerr-cat realized in a SNAIL (b). On the other hand, photons from the pumps at ω 1 and ω 2 such that | ω 1 − ω 2 | = | ω c, 1 − ω c, 2 |, convert a photon at ω c, 1 to ω c, 2 via four-wave mixing. Due to the fourth-order nonlinearity of the transmon, a photon each from the pumps at ω 3 and ω 4 such that ω 3 + ω 4 = ω c, 1 + ω c, 2, are consumed to generate two photons at ω c, 1 and ω c, 2. In this case, four pump tones can be used to generate the interaction. Alternatively, it is possible to engineer H ^ θ, ϕ using a transmon as shown in (b). This leads to an interaction of the form a ^ † b ^ † e i ϕ + H.c., where, again, the strength of the interaction and ϕ is set by the drive strength and phase, respectively.
Similarly, a single photon at ω 2 is consumed to create two photons, one at ω c, 1 and the other at ω c, 2. This leads to an interaction of the form a ^ † b ^ e i θ + H.c., where the strength of the interaction and θ depend on the strength of the microwave drive at ω 1 and its phase, respectively. Because of the third-order nonlinearity, a single photon at ω 1 is consumed to convert a photon at ω c, 1 to that at ω c, 2. Two microwave drives are applied at frequencies ω 1 and ω 2 so that ω 1 = | ω c, 1 − ω c, 2 | and ω 2 = ω c, 1 + ω c, 2. At an appropriate flux bias, the SNAIL exhibits a strong third-order nonlinearity. In (a), a superconducting nonlinear asymmetric inductive element (SNAIL) is used for the coupling. The two GKP resonators, shown in orange and blue, have frequencies ω c, 1 and ω c, 2, respectively. Illustration of how the Hamiltonian H ^ θ, ϕ ∝ e i θ a ^ b ^ † + e i ϕ a ^ b ^ + H.c., required for realization of the two-qubit Clifford gates C ¯ σ i σ j in Eq. ( 23), may be realized in a cQED architecture. We present our perspective on the challenges that lay ahead and discuss possible opportunities to meet them in order to harness the power of the GKP code, and build a large-scale fault-tolerant quantum computer. In this Perspective, we provide an overview of the GKP code, focusing on its realization in the superconducting circuit platform. These developments have laid a fertile ground for future research on scalable quantum computing with the GKP code.
#Quantum error correction with superconducting qubits code#
However, remarkable advances in quantum systems’ design and control over the last two decades has led to the realization of the GKP code in two very different experimental platforms: trapped ions and superconducting circuits.
Like many revolutionary ideas, at the time of its invention it was met with some reservation due to the complexity of realizing the quantum states defining the code.
There are strong reasons to believe that if implemented successfully, the GKP code could lead to a hardware-efficient solution for scalable quantum error correction. One of the earliest such continuous-variable error-correcting codes was introduced by Gottesman, Kitaev, and Preskill in 2001, now known as the GKP code after the name of its inventors. An example is the state of a quantum harmonic oscillator, which is characterized by its position and momentum. The principle behind QEC is simple yet remarkably powerful: quantum information is stored redundantly in nonlocal observables of a many-body quantum system, such that common errors, which damage parts of the system locally do not destroy the information globally.Īlthough the early quantum-error-correcting codes were designed with information stored in discrete variables, very soon the idea was applied to information stored in quantum states characterized by continuous variables. The introduction of quantum-error-correcting (QEC) codes in the early–mid 1990s transformed the field of quantum information science.
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