Superconducting quantum circuits are a promising hardware platform for realizing a fault-tolerant quantum computer. Accelerating progress in this field of research demands general approachesand computational tools to analyze and design more complex superconducting circuits. We develop a framework to systematically construct a superconducting quantum circuit’s quantized Hamiltonian from its physical description. As is often the case with quantum descriptions of multicoordinate systems, the complexity rises rapidly with the number of variables. Therefore, we introduce a set of coordinate transformations with which we can find bases to diagonalize the Hamiltonian efficiently. Furthermore, we broaden our framework’s scope to calculate the circuit’s key properties required for optimizing and discovering novel qubits. We implement the methods described in this work in an open-source Python package SQcircuit. In this manuscript, we introduce the reader to the SQcircuit environment and functionality. We show through a series of examples how to analyze a number of interesting quantum circuits and obtain features such as the spectrum, coherence times, transition matrix elements, coupling operators, and the phase coordinate representation of eigenfunctions.
We can encode a qubit in the energy levels of a quantum system. Relaxation and other dissipation processes lead to decay of the fidelity of this stored information. Is it possible topreserve the quantum information for a longer time by introducing additional drives and dissipation? The existence of autonomous quantum error correcting codes answers this question in the positive. Nonetheless, discovering these codes for a real physical system, i.e., finding the encoding and the associated driving fields and bath couplings, remains a challenge that has required intuition and inspiration to overcome. In this work, we develop and demonstrate a computational approach based on adjoint optimization for discovering autonomous quantum error correcting codes given a description of a physical system. We implement an optimizer that searches for a logical subspace and control parameters to better preserve quantum information. We demonstrate our method on a system of a harmonic oscillator coupled to a lossy qubit, and find that varying the Hamiltonian distance in Fock space — a proxy for the control hardware complexity — leads to discovery of different and new error correcting schemes. We discover what we call the 3‾√ code, realizable with a Hamiltonian distance d=2, and propose a hardware-efficient implementation based on superconducting circuits.
The evenly-spaced modes of an electromagnetic resonator are coupled to each other by appropriate time-modulation, leading to dynamics analogous to those of particles hopping betweendifferent sites of a lattice. This substitution of a real spatial dimension of a lattice with a „synthetic'“ dimension in frequency space greatly reduces the hardware complexity of an analog quantum simulator. Complex control and read-out of a highly multi-moded structure can thus be accomplished with very few physical control lines. We demonstrate this concept with microwave photons in a superconducting transmission line resonator by modulating the system parameters at frequencies near the resonator’s free spectral range and observing propagation of photon wavepackets in time domain. The linear propagation dynamics are equivalent to a tight-binding model, which we probe by measuring scattering parameters between frequency sites. We extract an approximate tight-binding dispersion relation for the synthetic lattice and initialize photon wavepackets with well-defined quasimomenta and group velocities. As an example application of this platform in simulating a physical system, we demonstrate Bloch oscillations associated with a particle in a periodic potential and subject to a constant external field. The simulated field strongly affects the photon dynamics despite photons having zero charge. Our observation of photon dynamics along a synthetic frequency dimension generalizes immediately to topological photonics and single-photon power levels, and expands the range of physical systems addressable by quantum simulation.