We introduce a general protocol for obtaining the charge basis density matrix of a superconducting quantum circuit. Inspired by cavity state tomography, our protocol combines Josephson-energy pulse sequences and projective charge-basis readout to access the off-diagonal elements of the density matrix, a scheme we thus dub charge basis tomography. We simulate the reconstruction of the ground state of a target transmon using the Aharonov-Casher effect in a probe qubit to realise projective readout and show the Hilbert-Schmidt distance can detect deviations from the correct model Hamiltonian. Unlocking this ability to validate models using the ground state sets the stage for using transmons to detect interacting and topological phases, particularly in materials where time-domain and spectroscopic probes can be limited by intrinsic noise.
We introduce a general protocol for obtaining the charge basis density matrix of a superconducting quantum circuit. Inspired by cavity state tomography, our protocol combines Josephson-energy pulse sequences and projective charge-basis readout to access the off-diagonal elements of the density matrix, a scheme we thus dub charge basis tomography. We simulate the reconstruction of the ground state of a target transmon using the Aharonov-Casher effect in a probe qubit to realise projective readout and show the Hilbert-Schmidt distance can detect deviations from the correct model Hamiltonian. Unlocking this ability to validate models using the ground state sets the stage for using transmons to detect interacting and topological phases, particularly in materials where time-domain and spectroscopic probes can be limited by intrinsic noise.
Abstract Quantum technologies rely heavily on accurate control and reliable readout of quantum systems. Current experiments are limited by numerous sources of noise that can only be partially captured by simple analytical models and additional characterization of the noise sources is required. We test the ability of readout error mitigation to correct noise found in systems composed of quantum two-level objects (qubits). To probe the limit of such methods, we designed a beyond-classical readout error mitigation protocol based on quantum state tomography (QST), which estimates the density matrix of a quantum system, and quantum detector tomography (QDT), which characterizes the measurement procedure. By treating readout error mitigation in the context of state tomography the method becomes largely readout mode-, architecture-, noise source-, and quantum state-independent. We implement this method on a superconducting qubit and evaluate the increase in reconstruction fidelity for QST. We characterize the performance of the method by varying important noise sources, such as suboptimal readout signal amplification, insufficient resonator photon population, off-resonant qubit drive, and effectively shortened T1 and T2 coherence. As a result, we identified noise sources for which readout error mitigation worked well, and observed decreases in readout infidelity by a factor of up to 30.
We propose a scheme to perform two-qubit state tomography by independently measuring the ensemble average of quantum states of two coupled superconducting transmon qubits. Each qubit is capacitively coupled to its own readout cavity and can be measured separately. In order to obtain the density matrix of a two-qubit state, we apply four two-qubit unitary operations to the initial state and measure the corresponding qubit states, from which the elements of the two-qubit density matrix can be extracted. By using this scheme, we measure the entangled two qubits with high fidelity.
We solve the problem of an interacting quantum dot embedded in a Josephson junction between two superconductors with finite charging energy described by the transmon (Cooper pair box) Hamiltonian. The approach is based on the flat-band approximation of the Richardson model, which reduces the Hilbert space to the point where exact diagonalization is possible while retaining all states that are necessary to describe the low-energy phenomena. The presented method accounts for the physics of the quantum dot, the Josephson effect, and the Coulomb repulsion (charging energy) at the same level. In particular, it captures the quantum fluctuations of the superconducting phase as well as the coupling between the superconducting phase and the quantum dot (spin) degrees of freedom. The method can be directly applied for modeling Andreev spin qubits embedded in transmon circuits in all parameter regimes, for describing time-dependent processes, and for the calculation of transition matrix elements for microwave-driven transmon, spin-flip, and mixed transitions that involve coupling to charge or current degrees of freedom.
Nonlinear superconducting circuits are a leading candidate to implement quantum information tasks beyond the reach of classical processors. One such task, quantum simulation, involves using a controllable quantum system to study the dynamics of another. In this thesis, we present a series of experiments using small systems of superconducting circuits to perform various quantum tasks relevant to quantum simulation.In the first experiment, we design and build a circuit comprising three transmon qubits whose dynamics mirror those of interacting bosonic particles on a lattice, described by the Bose-Hubbard Hamiltonian. We verify the predictions of the Bose-Hubbard model for this system spectroscopically and then use time-domain measurements to study the decoherence processes affecting the qubits. Using a Raman process, we engineer artificial decay dynamics in the system which allow us to prepare and stabilize otherwise inaccessible states of the transmon array.The second experimental demonstration concerns topological quantum matter. Certain quantum systems are characterized by quantities known as topological invariants, which are robust to local perturbations. These topological invariants are challenging to measure in naturally-occurring systems. We engineer an artificial system, comprising a transmon qubit coupled to a high-Q cavity, capable of undergoing a protocol known as the quantum walk, which also exhibits topological invariants. By using particular non-classical state of the cavity, we modify the quantum walk protocol to make the topological invariant directly accessible.Finally, we implement a hybrid quantum-classical algorithm—known as the variational quantum eigensolver—in a two-transmon quantum processor. As a proof of principle demonstration, we compute the ground-state and low-lying excited state energies of the hydrogen molecule as a function of nuclear separation. We show that an extension of the basic algorithm, known as the quantum subspace expansion, allows for the mitigation of errors caused by decoherence processes affecting the quantum processor.
We theoretically study a scheme to distinguish the two ground states of a one-dimensional topological superconductor, which could serve as a basis for the readout of Majorana qubits. The scheme is based on parity-to-charge conversion, i.e., the ground-state parity of the superconductor is converted to the charge occupation on a tunnel-coupled auxiliary quantum dot. We describe how certain error mechanisms degrade the quality of the parity-to-charge conversion process. We consider (i) leakage due to a strong readout tunnel pulse, (ii) incomplete charge Rabi oscillations due to slow charge noise, and (iii) charge relaxation due to phonon emission and absorption. To describe these effects, we use simple model Hamiltonians based on the ideal Kitaev chain, and draw conclusions to generic one-dimensional topological superconductors wherever possible. In general, the effects of the error mechanisms can be minimized by choosing a smooth shape and an optimal strength for the readout tunnel pulse. In a case study based on InAs heterostructure device parameters, we estimate that the parity-to-charge conversion error is mainly due to slow charge noise for weak tunnel pulses and leakage for strong tunnel pulses.
A protocol to test cross correlations between charge and critical-current noise in the small superconducting contacts of an asymmetric Cooper-pair transistor coupled to a phase qubit is presented. The superconducting circuit behaves as a tunable four-level quantum system that can be prepared in two different configurations where cross-correlation terms are, respectively, absent or present and therefore, in principle, detectable. The measurements of the cross correlations are performed either through the escape probability of the dc superconducting quantum interference device or a quantum-state tomography of a few elements of the reduced density matrix of the four-level quantum system.