##TOP## Free Online Quantum Circuit Simulator

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QuTiP is open-source software for simulating the dynamics of open quantum systems. The QuTiP library depends on the excellent Numpy, Scipy, and Cython numerical packages. In addition, graphical output is provided by Matplotlib. QuTiP aims to provide user-friendly and efficient numerical simulations of a wide variety of Hamiltonians, including those with arbitrary time-dependence, commonly found in a wide range of physics applications such as quantum optics, trapped ions, superconducting circuits, and quantum nanomechanical resonators. QuTiP is freely available for use and/or modification on all major platforms such as Linux, Mac OSX, and Windows*. Being free of any licensing fees, QuTiP is ideal for exploring quantum mechanics and dynamics in the classroom.

TensorFlow Quantum focuses on quantum data and building hybrid quantum-classical models. It integrates quantum computing algorithms and logic designed in Cirq, and provides quantum computing primitives compatible with existing TensorFlow APIs, along with high-performance quantum circuit simulators. Read more in the TensorFlow Quantum white paper.

The variational quantum eigensolver is currently the flagship algorithm for solving electronic structure problems on near-term quantum computers. The algorithm involves implementing a sequence of parameterized gates on quantum hardware to generate a target quantum state, and then measuring the molecular energy. Due to finite coherence times and gate errors, the number of gates that can be implemented remains limited. In this work, we propose an alternative algorithm where device-level pulse shapes are variationally optimized for the state preparation rather than using an abstract-level quantum circuit. In doing so, the coherence time required for the state preparation is drastically reduced. We numerically demonstrate this by directly optimizing pulse shapes which accurately model the dissociation of H2 and HeH+, and we compute the ground state energy for LiH with four transmons where we see reductions in state preparation times of roughly three orders of magnitude compared to gate-based strategies.

In order to turn this strategy into an algorithm, one needs a procedure for realizing the target molecular wavefunction on the QPU. As the leading quantum algorithm for molecular simulation on near-term devices, the variational quantum eigensolver (VQE)14 provides an efficient procedure for this purpose. In VQE, one defines a parameterized quantum circuit comprised of tunable gates, and then optimizes these gates using the variational principle, minimizing the energy of the molecular Hamiltonian. This parameterized quantum circuit (referred to as an ansatz) defines the variational flexibility (and thus the subspace reachable on the QPU) of the algorithm.

State-preparation circuits with more parameters generally have more variational flexibility, but come with the cost of having deeper quantum circuits and more difficult optimization. This cost can be significant. Current and near-term quantum computers are classified as noisy intermediate scale quantum (NISQ) devices due to the presence of short coherence times, system noise, and frequent gate errors15. Because each gate has limited fidelity, the success probability for a sequence of gates decreases exponentially with circuit depth. Even if gates could reach unit fidelity, the finite coherence times of a NISQ device still limits the number of gates that one can apply in a circuit which, in turn, limits the accuracy of the molecular VQE simulation. Although VQE is relatively robust in the presence of noise and errors in certain cases16, the critical limitation preventing larger scale experiments is the accurate implementation of deep circuits. The goal of finding parameterized circuits which minimize the circuit depth and maximize the accuracy has led to a number of approaches such as hardware efficient ansätze17, physically motivated fixed ansätze18,19,20,21,22, and adaptive ansätze 23,24,25.

In this paper, we explore the possibility of performing gate-free VQE simulations by replacing the parameterized quantum circuit with a direct optimization of the laboratory-frame analog control settings. In the following sections, we argue that quantum control techniques are likely to be better suited for fast VQE state preparation than the more conventional circuit-based approaches on NISQ devices. We first provide a detailed overview of circuit-based VQE, then introduce our proposed strategy, ctrl-VQE, then discuss initial results along with a strategy to avoid over-parameterization, and finally compare to gate-based ansätze. Several technical aspects, numerical results, and additional discussion are provided in the Supplementary Methods and Supplementary Discussion.

Although several ansätze have been proposed to achieve shorter circuits, even the most compact approaches involve too many gates to implement on current hardware for all but the smallest molecules. In order to reduce the time spent during the state-preparation stage, and thus the coherence time demands on the hardware, circuit compilation techniques have been designed to take a given quantum circuit and execute it either with fewer gates or by reoptimizing groups of gates for faster execution.

To execute the gate-based VQE (described in Section III A) experimentally, the gates in a circuit are compiled to sequences of analog control pulses using a look-up table that maps each elementary gate to the associated predefined analog pulse. The sequence of control pulses corresponding to the elementary gates in the quantum circuit are then simply collected and appropriately scheduled to be executed on the hardware. As such the compilation is essentially instantaneous, making the gate-based compilation technique well suited for VQE algorithms where numerous iterations are performed.

The GRAPE compilation technique employs an optimal control routine which compiles to machine-level sequences of analog pulses for a target quantum circuit. This is achieved by manipulating the sets of time-discrete control fields that are executed on the quantum system. The control fields in the optimal routine are updated using gradients (see Boutin et al.44 for use of analytical gradients) of the cost function with respect to the control fields. GRAPE compilation achieves up to 5-fold speedups compared to standard gate-based compilation. However, such speedup comes with a substantial computational cost. This amounts to long compilation latency, making this approach impractical for VQE algorithms in which multiple iterations of circuit-parameter optimizations are performed. The GRAPE-based compilation also suffers from limitations in the circuit sizes it can handle45,46,47.

On the other hand, partial compilation techniques achieve significant pulse speedups by leveraging the fast compilation of standard gate-based techniques together with the pulse speedups of GRAPE-based compilations. Two flavors of such an approach were reported in Gokhale et al.43. Both divide the whole circuit into blocks of subcircuits. In the so-called strict partial compilation, the structure of quantum circuits used in quantum variational algorithms are exploited to only perform GRAPE-based compilation on fixed subcircuits that are independent of the circuit parametrization. The optimal pulses using the GRAPE compilation techniques for the fixed blocks are pre-computed and simply concatenated with the control pulses from the gate-based compilations for the remaining blocks of the circuit. Thus, the compilation speed is comparable to the gate-based compilations in each iteration, but this method also takes advantage of pulse speedups from GRAPE-based compilations. As one may expect, the pulse speedups heavily depend on the circuit depth of the fixed blocks.

In summary, we have presented an alternative new variational quantum algorithm which is fundamentally different from the existing quantum algorithms for molecular simulation. The quantum circuit used for state preparation in standard variational algorithms is entirely replaced by a hardware-level control routine with optimized pulse shapes to directly drive the qubit system. As such, we demonstrate that VQE applications do not need to be confined to two level systems (qubits), which are required for general quantum computing. This opens a possibility of faster ansatz preparation within a time window defined by coherence times of NISQ devices.

The VQE algorithm aims to leverage classical resources to reduce the circuit depth required for molecular simulation. The algorithm finds optimal rotation angles for a parameterized quantum circuit of fixed depth by variationally minimizing the energy of a target molecule, which is obtained by repeated state preparation and measurement cycles. In order to account for the distinguishability of qubits, we start by transforming the second quantized electronic Hamiltonian into an equivalent form involving non-local strings of Pauli spin operators, \({\hat{o}}_{i}\):

In this section, we present an alternative to the gate-based VQE algorithm, replacing the parameterized state-preparation circuit with a parameterized laboratory-frame pulse representation, which is optimized in an analogous manner, but with the benefit of a much faster state preparation, opening up the possibility of more accurate simulations on NISQ devices. All other aspects of VQE (i.e., measurement protocols) are essentially the same. Using the molecular energy as the objective function to minimize, the pulse parameters are optimized using the variational principle. This general strategy, which we refer to as control variational quantum eigensolver (ctrl-VQE), is outlined as follows:

The ansatz above is completely (and solely) determined by the device and controls, granting enormous flexibility to the ansatz. In fact, any digital quantum circuit ansatz can be compiled into the form above. As such, the ansatz in Eq. (9) does not intrinsically possess any limitation on its potential accuracy beyond the fundamental limitations imposed by quantum speed limits40. However, this additional flexibility can make optimization more difficult. 2b1af7f3a8