The required precision to perform quantum simulations beyond the capabilities of classical computers imposes major experimental and theoretical challenges. Here, we develop a characterization technique to benchmark the implementation precision of a specific quantum simulation task. We infer all parameters of the bosonic Hamiltonian that governs the dynamics of excitations in a two-dimensional grid of nearest-neighbour coupled superconducting qubits. We devise a robust algorithm for identification of Hamiltonian parameters from measured times series of the expectation values of single-mode canonical coordinates. Using super-resolution and denoising methods, we first extract eigenfrequencies of the governing Hamiltonian from the complex time domain measurement; next, we recover the eigenvectors of the Hamiltonian via constrained manifold optimization over the orthogonal group. For five and six coupled qubits, we identify Hamiltonian parameters with sub-MHz precision and construct a spatial implementation error map for a grid of 27 qubits. Our approach enables us to distinguish and quantify the effects of state preparation and measurement errors and show that they are the dominant sources of errors in the implementation. Our results quantify the implementation accuracy of analog dynamics and introduce a diagnostic toolkit for understanding, calibrating, and improving analog quantum processors.
The required precision to perform quantum simulations beyond the capabilities of classical computers imposes major experimental and theoretical challenges. Here, we develop a characterization technique to benchmark the implementation precision of a specific quantum simulation task. We infer all parameters of the bosonic Hamiltonian that governs the dynamics of excitations in a two-dimensional grid of nearest-neighbour coupled superconducting qubits. We devise a robust algorithm for identification of Hamiltonian parameters from measured times series of the expectation values of single-mode canonical coordinates. Using super-resolution and denoising methods, we first extract eigenfrequencies of the governing Hamiltonian from the complex time domain measurement; next, we recover the eigenvectors of the Hamiltonian via constrained manifold optimization over the orthogonal group. For five and six coupled qubits, we identify Hamiltonian parameters with sub-MHz precision and construct a spatial implementation error map for a grid of 27 qubits. Our approach enables us to distinguish and quantify the effects of state preparation and measurement errors and show that they are the dominant sources of errors in the implementation. Our results quantify the implementation accuracy of analog dynamics and introduce a diagnostic toolkit for understanding, calibrating, and improving analog quantum processors.
The required precision to perform quantum simulations beyond the capabilities of classical computers imposes major experimental and theoretical challenges. The key to solving these issues are precise means of characterizing analog quantum simulators. Here, we robustly estimate the free Hamiltonian parameters of bosonic excitations in a superconducting-qubit analog quantum simulator from measured time-series of single-mode canonical coordinates. We achieve high levels of precision in estimating the Hamiltonian parameters by exploiting a priori knowledge, making it robust against noise and state-preparation and measurement (SPAM) errors. Importantly, we are also able to obtain tomographic information about those SPAM errors from the same data, crucial for the experimental applicability of Hamiltonian learning in dynamical quantum-quench experiments. Our learning algorithm is scalable both in terms of the required amounts of data and post-processing. To achieve this, we develop a new super-resolution technique coined tensorESPRIT for frequency extraction from matrix time-series. The algorithm then combines tensorESPRIT with constrained manifold optimization for the eigenspace reconstruction with pre- and post-processing stages. For up to 14 coupled superconducting qubits on two Sycamore processors, we identify the Hamiltonian parameters -- verifying the implementation on one of them up to sub-MHz precision -- and construct a spatial implementation error map for a grid of 27 qubits. Our results constitute an accurate implementation of a dynamical quantum simulation that is characterized using a new diagnostic toolkit for understanding, calibrating, and improving analog quantum processors.
Abstract Precise means of characterizing analog quantum simulators are key to developing quantum simulators capable of beyond-classical computations. Here, we precisely estimate the free Hamiltonian parameters of a superconducting-qubit analog quantum simulator from measured time-series data on up to 14 qubits. To achieve this, we develop a scalable Hamiltonian learning algorithm that is robust against state-preparation and measurement (SPAM) errors and yields tomographic information about those SPAM errors. The key subroutines are a novel super-resolution technique for frequency extraction from matrix time-series, tensorESPRIT, and constrained manifold optimization. Our learning results verify the Hamiltonian dynamics on a Sycamore processor up to sub-MHz accuracy, and allow us to construct a spatial implementation error map for a grid of 27 qubits. Our results constitute an accurate implementation of a dynamical quantum simulation that is precisely characterized using a new diagnostic toolkit for understanding, calibrating, and improving analog quantum processors.
Noise characterisation not only enables error mitigation but also provides valuable information about the intrinsic performance of a quantum device, identifying dominant error contributions and bounding noise stability well beyond the aggregated calibration data. Deploying such methods on a specific hardware topology must combine efficiency and accuracy, assess noise stability, and validate the reconstructed noise model. Here we develop and implement these methods on a superconducting processor with square-lattice topology. First, we revise the cycle-benchmarking protocol and reduce the bias arising from unlearnable degrees of freedom via a regularised cost function. Second, we extend a previously developed tensor-network tomography method to match the device layout and show how its accuracy can be improved by repeated layer execution. Both methods allow approximate sublayer noise models to be extracted from parent ones. Using noise characterisation as a tool, we map the noise structure, track drift, and validate the adequacy of the learnt noise models against experimentally observed noisy estimations in benchmark physical and chemistry circuits. These results inform further improvements in the underlying technology and provide estimates of the size of noise-mitigated circuits that the device can support.
As progress is made towards the first generation of error-corrected quantum computers, robust characterization and validation protocols are required to assess the noise environments of physical quantum processors. While standard coherence metrics and characterization protocols such as T1 and T2, process tomography, and randomized benchmarking are now ubiquitous, these techniques provide only partial information about the dynamic multiqubit loss channels responsible for processor errors, which can be described more fully by a Lindblad operator in the master equation formalism. Here, we introduce and experimentally demonstrate Lindblad tomography, a hardware-agnostic characterization protocol for tomographically reconstructing the Hamiltonian and Lindblad operators of a quantum noise environment from an ensemble of time-domain measurements. Performing Lindblad tomography on a small superconducting quantum processor, we show that this technique naturally builds on standard process tomography and T1/T2 measurement protocols, characterizes and accounts for state-preparation and measurement errors, and allows one to place bounds on the fit to a Markovian model. Comparing the results of single- and two-qubit measurements on a superconducting quantum processor, we demonstrate that Lindblad tomography can also be used to identify and quantify sources of crosstalk on quantum processors, such as the presence of always-on qubit-qubit interactions.
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 demonstrate a method to characterize the general Heisenberg Hamiltonian with non-uniform couplings by mapping the entanglement it generates as a function of time. Identification of the Hamiltonian in this way is possible as the coefficients of each operator control the oscillation frequencies of the entanglement function. The number of measurements required to achieve a given precision in the Hamiltonian parameters is determined and an efficient measurement strategy designed. We derive the relationship between the number of measurements, the resulting precision and the ultimate discrete error probability generated by a systematic mis-characterization. This has important implications when implementing two-qubit gates for fault-tolerant quantum computation.