Harnessing the full power of nascent quantum processors requires the efficient management of a limited number of quantum bits with finite lifetime. Hybrid algorithms leveraging classicalresources have demonstrated promising initial results in the efficient calculation of Hamiltonian ground states–an important eigenvalue problem in the physical sciences that is often classically intractable. In these protocols, a Hamiltonian is parsed and evaluated term-wise with a shallow quantum circuit, and the resulting energy minimized using classical resources. This reduces the number of consecutive logical operations that must be performed on the quantum hardware before the onset of decoherence. We demonstrate a complete implementation of the Variational Quantum Eigensolver (VQE), augmented with a novel Quantum Subspace Expansion, to calculate the complete energy spectrum of the H2 molecule with near chemical accuracy. The QSE also enables the mitigation of incoherent errors, potentially allowing the implementation of larger-scale algorithms without complex quantum error correction techniques.
The direct measurement of topological invariants in both engineered and naturally occurring quantum materials is a key step in classifying quantum phases of matter. Here we motivatea toolbox based on time-dependent quantum walks as a method to digitally simulate single-particle topological band structures. Using a superconducting qubit dispersively coupled to a microwave cavity, we implement two classes of split-step quantum walks and directly measure the topological invariant (winding number) associated with each. The measurement relies upon interference between two components of a cavity Schr\“odinger cat state and highlights a novel refocusing technique which allows for the direct implementation of a digital version of Bloch oscillations. Our scheme can readily be extended to higher dimensions, whereby quantum walk-based simulations can probe topological phases ranging from the quantum spin Hall effect to the Hopf insulator.
The topology of a single-particle band structure plays a fundamental role in understanding a multitude of physical phenomena. Motivated by the connection between quantum walks and suchtopological band structures, we demonstrate that a simple time-dependent, Bloch-oscillating quantum walk enables the direct measurement of topological invariants. We consider two classes of one-dimensional quantum walks and connect the global phase imprinted on the walker with its refocusing behavior. By disentangling the dynamical and geometric contributions to this phase we describe a general strategy to measure the topological invariant in these quantum walks. As an example, we propose an experimental protocol in a circuit QED architecture where a superconducting transmon qubit plays the role of the coin, while the quantum walk takes place in the phase space of a cavity.
In quantum mechanics, measurement restores a classical notion of reality via collapse of the wavefunction, which yields a precisely defined outcome. On the other hand, the Heisenberguncertainty principle dictates that incompatible observables, such as position and momentum, cannot both take on arbitrarily precise values. But how does a wavefunction evolve when two such quantities are probed simultaneously, and how does the uncertainty principle dynamically inhibit precise measurement outcomes? To realize this unexplored regime, we simultaneously apply two continuous quantum non-demolition probes of non-commuting observables on a superconducting qubit. We achieve this capability by developing a novel measurement scheme that allows us to control the axes of multiple readout channels. We show that the uncertainty principle directly governs the dynamics of the state, and consequently standard wavefunction collapse is replaced by a persistent diffusion that exhibits several distinct regimes. Although evolution of the state now differs drastically from that of a conventional measurement, information about both non-commuting observables is extracted by keeping track of the time ordering of the measurement record, enabling quantum state tomography without alternating measurements. Our work creates new capabilities for quantum control, including rapid state purification, adaptive measurement, measurement-based state steering and continuous quantum error correction. As physical quantum systems interact with their environments via non-commuting degrees of freedom, our work offers a new, more natural approach to experimentally study contemporary quantum foundations.