Implementing and characterizing precise multi-qubit measurements

There are two general requirements to harness the computational power of quantum mechanics: the ability to manipulate the evolution of an isolated system and the ability to faithfully extract information from it. Quantum error correction and simulation often make a more exacting demand: the ability to perform non-destructive measurements of specific correlations within that system. We realize such measurements by employing a protocol adapted from [S. Nigg and S. M. Girvin, Phys. Rev. Lett. 110, 243604 (2013)], enabling real-time selection of arbitrary register-wide Pauli operators. Our implementation consists of a simple circuit quantum electrodynamics (cQED) module of four highly-coherent 3D transmon qubits, collectively coupled to a high-Q superconducting microwave cavity. As a demonstration, we enact all seven nontrivial subset-parity measurements on our three-qubit register. For each we fully characterize the realized measurement by analyzing the detector (observable operators) via quantum detector tomography and by analyzing the quantum back-action via conditioned process tomography. No single quantity completely encapsulates the performance of a measurement, and standard figures of merit have not yet emerged. Accordingly, we consider several new fidelity measures for both the detector and the complete measurement process. We measure all of these quantities and report high fidelities, indicating that we are measuring the desired quantities precisely and that the measurements are highly non-demolition. We further show that both results are improved significantly by an additional error-heralding measurement. The analyses presented here form a useful basis for the future characterization and validation of quantum measurements, anticipating the demands of emerging quantum technologies.Comment: 10 pages, 5 figures, plus supplemen

Logic gates at the surface code threshold: Superconducting qubits poised for fault-tolerant quantum computing

A quantum computer can solve hard problems - such as prime factoring, database searching, and quantum simulation - at the cost of needing to protect fragile quantum states from error. Quantum error correction provides this protection, by distributing a logical state among many physical qubits via quantum entanglement. Superconductivity is an appealing platform, as it allows for constructing large quantum circuits, and is compatible with microfabrication. For superconducting qubits the surface code is a natural choice for error correction, as it uses only nearest-neighbour coupling and rapidly-cycled entangling gates. The gate fidelity requirements are modest: The per-step fidelity threshold is only about 99%. Here, we demonstrate a universal set of logic gates in a superconducting multi-qubit processor, achieving an average single-qubit gate fidelity of 99.92% and a two-qubit gate fidelity up to 99.4%. This places Josephson quantum computing at the fault-tolerant threshold for surface code error correction. Our quantum processor is a first step towards the surface code, using five qubits arranged in a linear array with nearest-neighbour coupling. As a further demonstration, we construct a five-qubit Greenberger-Horne-Zeilinger (GHZ) state using the complete circuit and full set of gates. The results demonstrate that Josephson quantum computing is a high-fidelity technology, with a clear path to scaling up to large-scale, fault-tolerant quantum circuits.Comment: 15 pages, 13 figures, including supplementary materia

Implementation of variational quantum algorithms on superconducting qudits

Quantum computing is considered an emerging technology with promising applications in chemistry, materials, medicine, and cryptography. Superconducting circuits are a leading candidate hardware platform for the realisation of quantum computing, and superconducting devices have now been demonstrated at a scale of hundreds of qubits. Further scale-up faces challenges in wiring, frequency crowding, and the high cost of control electronics. Complementary to increasing the number of qubits, using qutrits (3-level systems) or qudits (d-level systems, d>3) as the basic building block for quantum processors can also increase their computational capability. A commonly used superconducting qubit design, the transmon, has more than two levels. It is a good candidate for a qutrit or qudit processor. Variational quantum algorithms are a type of quantum algorithm that can be implemented on near-term devices. They have been proposed to have a higher tolerance to noise in near-term devices, making them promising for near-term applications of quantum computing. The difference between qubits and qudits makes it non-trivial to translate a variational algorithm designed for qubits onto a qudit quantum processor. The algorithm needs to be either rewritten into a qudit version or an emulator needs to be developed to emulate a qubit processor with a qudit processor. This thesis describes research on the implementation of variational quantum algorithms, with a particular focus on utilising more than two computational levels of transmons. The work comprises building a two-qubit transmon device and a multi-level transmon device that is used as a qutrit or a qudit (d = 4). We fully benchmarked the two-qubit and the single qudit devices with randomised benchmarking and gate-set tomography, and found good agreement between the two approaches. The qutrit Hadamard gate is reported to have an infidelity of 3.22 ± 0.11 × 10−3, which is comparable to state-of-the-art results. We use the qudit to implement a two-qubit emulator and report that the two-qubit Clifford gate randomised benchmarking result on the emulator (infidelity 9.5 ± 0.7 × 10−2) is worse than the physical two-qubit (infidelity 4.0 ± 0.3 × 10−2) result. We also implemented active reset for the qudit transmon to demonstrate preparing high-fidelity initial states with active feedback. We found the initial state fidelity improved from 0.900 ± 0.011 to 0.9932 ± 0.0013 from gate set tomography. We finally utilised the single qudit device to implement quantum algorithms. First, a single qutrit classifier for the iris dataset was implemented. We report a successful demonstration of the iris classifier, which yields the training accuracy of the qutrit classifier as 0.96 ± 0.03 and the testing accuracy as 0.94 ± 0.04 among multiple trials. Second, we implemented a two-qubit emulator with a 4-level qudit and used the emulator to demonstrate a variational quantum eigensolver for hydrogen molecules. The solved energy versus the hydrogen bond distance is within 1.5 × 10−2 Hartree, below the chemical accuracy threshold. From the characterisation, benchmarking results, and successful demonstration of two quantum algorithms, we conclude that higher levels of a transmon can be used to increase the size of the Hilbert space used for quantum computation with minimal extra cost

Optimal synchronization deep in the quantum regime: resource and fundamental limit

We develop an analytical framework to study the synchronization of a quantum self-sustained oscillator to an external signal. Our unified description allows us to identify the resource on which quantum synchronization relies, and to compare quantitatively the synchronization behavior of different limit cycles and signals. We focus on the most elementary quantum system that is able to host a self-sustained oscillation, namely a single spin 1. Despite the spin having no classical analogue, we first show that it can realize the van der Pol limit cycle deep in the quantum regime, which allows us to provide an analytical understanding to recently reported numerical results. Moving on to the equatorial limit cycle, we then reveal the existence of an interference-based quantum synchronization blockade and extend the classical Arnold tongue to a snake-like split tongue. Finally, we derive the maximum synchronization that can be achieved in the spin-1 system, and construct a limit cycle that reaches this fundamental limit asymptotically.Comment: 15 pages, 9 figures, equivalent to published versio

Phase Retrieval Using Unitary 2-Designs

We consider a variant of the phase retrieval problem, where vectors are replaced by unitary matrices, i.e., the unknown signal is a unitary matrix U, and the measurements consist of squared inner products |Tr(C*U)|^2 with unitary matrices C that are chosen by the observer. This problem has applications to quantum process tomography, when the unknown process is a unitary operation. We show that PhaseLift, a convex programming algorithm for phase retrieval, can be adapted to this matrix setting, using measurements that are sampled from unitary 4- and 2-designs. In the case of unitary 4-design measurements, we show that PhaseLift can reconstruct all unitary matrices, using a near-optimal number of measurements. This extends previous work on PhaseLift using spherical 4-designs. In the case of unitary 2-design measurements, we show that PhaseLift still works pretty well on average: it recovers almost all signals, up to a constant additive error, using a near-optimal number of measurements. These 2-design measurements are convenient for quantum process tomography, as they can be implemented via randomized benchmarking techniques. This is the first positive result on PhaseLift using 2-designs.Comment: 21 pages; v3: minor revisions, to appear at SampTA 2017; v2: rewritten to focus on phase retrieval, with new title, improved error bounds, and numerics; v1: original version, titled "Quantum Compressed Sensing Using 2-Designs

Quantum process tomography of molecular dimers from two-dimensional electronic spectroscopy I: General theory and application to homodimers

Is it possible to infer the time evolving quantum state of a multichromophoric system from a sequence of two-dimensional electronic spectra (2D-ES) as a function of waiting time? Here we provide a positive answer for a tractable model system: a coupled dimer. After exhaustively enumerating the Liouville pathways associated to each peak in the 2D-ES, we argue that by judiciously combining the information from a series of experiments varying the polarization and frequency components of the pulses, detailed information at the amplitude level about the input and output quantum states at the waiting time can be obtained. This possibility yields a quantum process tomography (QPT) of the single-exciton manifold, which completely characterizes the open quantum system dynamics through the reconstruction of the process matrix. This is the first of a series of two articles. In this manuscript, we specialize our results to the case of a homodimer, where we prove that signals stemming from coherence to population transfer and viceversa vanish upon isotropic averaging, and therefore, only a partial QPT is possible in this case. However, this fact simplifies the spectra, and it follows that only two polarization controlled experiments (and no pulse-shaping requirements) suffice to yield the elements of the process matrix which survive under isotropic averaging. The angle between the two site transition dipole moments is self-consistently obtained from the 2D-ES. Model calculations are presented, as well as an error analysis in terms of the angle between the dipoles and peak overlap. In the second article accompanying this study, we numerically exemplify the theory for heterodimers and carry out a detailed error analysis for such case. This investigation provides an important benchmark for more complex proposals of quantum process tomography (QPT) via multidimensional spectroscopic experiments