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Register a new userA Classical Algorithm for Quantum $textsf{SU}(2)$ Schur Sampling
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Many quantum algorithms can be represented in a form of a classical circuit positioned between quantum Fourier transformations. Motivated by the search for new quantum algorithms, we turn to circuits where the latter transformation is replaced by the $textsf{SU}(2)$ quantum Schur Transform -- a global transformation which maps the computational basis to a basis defined by angular momenta. We show that the output distributions of these circuits can be approximately classically sampled in polynomial time if they are sufficiently close to being sparse, thus isolating a regime in which these Quantum $textsf{SU}(2)$ Schur Circuits could lead to algorithms with exponential computational advantage. Our work is primarily motivated by a conjecture that underpinned the hardness of Permutational Quantum Computing, a restricted quantum computational model that has the above circuit structure in one of its computationally interesting regimes. The conjecture stated that approximating transition amplitudes of Permutational Quantum Computing model to inverse polynomial precision on a classical computer is computationally hard. We disprove the extended version of this conjecture -- even in the case when the hardness of approximation originated from a difficulty of finding the large elements in the output probability distributions. Finally, we present some evidence that output of the above Permutational Quantum Computing circuits could be efficiently approximately sampled from on a classical computer.
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Permutational Quantum Computing (PQC) [emph{Quantum~Info.~Comput.}, textbf{10}, 470--497, (2010)] is a natural quantum computational model conjectured to capture non-classical aspects of quantum computation. An argument backing this conjecture was the observation that there was no efficient classical algorithm for estimation of matrix elements of the $S_n$ irreducible representation matrices in the Youngs orthogonal form, which correspond to transition amplitudes of a broad class of PQC circuits. This problem can be solved with a PQC machine in polynomial time, but no efficient classical algorithm for the problem was previously known. Here we give a classical algorithm that efficiently approximates the transition amplitudes up to polynomial additive precision and hence solves this problem. We further extend our discussion to show that transition amplitudes of a broader class of quantum circuits -- the Quantum Schur Sampling circuits -- can be also efficiently estimated classically.
Hamiltonian learning is crucial to the certification of quantum devices and quantum simulators. In this paper, we propose a hybrid quantum-classical Hamiltonian learning algorithm to find the coefficients of the Pauli operator components of the Hamiltonian. Its main subroutine is the practical log-partition function estimation algorithm, which is based on the minimization of the free energy of the system. Concretely, we devise a stochastic variational quantum eigensolver (SVQE) to diagonalize the Hamiltonians and then exploit the obtained eigenvalues to compute the free energys global minimum using convex optimization. Our approach not only avoids the challenge of estimating von Neumann entropy in free energy minimization, but also reduces the quantum resources via importance sampling in Hamiltonian diagonalization, facilitating the implementation of our method on near-term quantum devices. Finally, we demonstrate our approachs validity by conducting numerical experiments with Hamiltonians of interest in quantum many-body physics.
We show that low-depth random quantum circuits can be efficiently simulated by a quantum teleportation-inspired algorithm. By using logical qubits to redirect and teleport the quantum information in quantum circuits, the original circuits can be renormalized to new circuits with a smaller number of logical qubits. We demonstrate the algorithm to simulate several random quantum circuits, including 1D-chain 1000-qubit 42-depth, 2D-grid 125*8-qubit 42-depth and 2D-Bristlecone 72-qubit 32-depth circuits. Our results present a memory-efficient method with a clear physical picture to simulate low-depth random quantum circuits.
We realize, for the first time, a non-Abelian gauge theory with both gauge and matter fields on a quantum computer. This enables the observation of hadrons and the calculation of their associated masses. The SU(2) gauge group considered here represents an important first step towards ultimately studying quantum chromodynamics, the theory that describes the properties of protons, neutrons and other hadrons. Quantum computers are able to create important new opportunities for ongoing essential research on gauge theories by providing simulations that are unattainable on classical computers. Our calculations on an IBM superconducting platform utilize a variational quantum eigensolver to study both meson and baryon states, hadrons which have never been seen in a non-Abelian simulation on a quantum computer. We develop a resource-efficient approach that not only allows the implementation of a full SU(2) gauge theory on present-day quantum hardware, but further lays out the premises for future quantum simulations that will address currently unanswered questions in particle and nuclear physics.
Improving the efficiency and accuracy of energy calculations has been of significant and continued interest in the area of materials informatics, a field that applies machine learning techniques to computational materials data. Here, we present a heuristic quantum-classical algorithm to efficiently model and predict the energies of substitutionally disordered binary crystalline materials. Specifically, a quantum circuit that scales linearly in the number of lattice sites is designed and trained to predict the energies of quantum chemical simulations in an exponentially-scaling feature space. This circuit is trained by classical supervised-learning using data obtained from classically-computed quantum chemical simulations. As a part of the training process, we introduce a sub-routine that is able to detect and rectify anomalies in the input data. The algorithm is demonstrated on the complex layer-structured of Li-cobaltate system, a widely-used Li-ion battery cathode material component. Our results shows that the proposed quantum circuit model presents a suitable choice for modelling the energies obtained from such quantum mechanical systems. Furthermore, analysis of the anomalous data provides important insights into the thermodynamic properties of the systems studied.
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