In this paper, we develop a Lie group theoretic approach for parametric representation of unitary matrices. This leads to develop a quantum neural network framework for quantum circuit approximation of multi-qubit unitary gates. Layers of the neural networks are defined by product of exponential of certain elements of the Standard Recursive Block Basis, which we introduce as an alternative to Pauli string basis for matrix algebra of complex matrices of order
2n. The recursive construction of the neural networks implies that the quantum circuit approximation is scalable i.e. quantum circuit for an
(n+1)-qubit unitary can be constructed from the circuit of
n-qubit system by adding a few CNOT gates and single-qubit gates.
In this work, scalable quantum neural networks are introduced to approximate unitary evolutions through the Standard Recursive Block Basis (SRBB) and, subsequently, redesigned with a reduced number of CNOTs. This algebraic approach to the problem of unitary synthesis exploits Lie algebras and their topological features to obtain a continuous parameterization of unitary operators. First, we implemented the recursive construction of the SRBB to provide a starting tool for quickly verifying the mathematical properties needed to guarantee the original scalability scheme, already known to the literature only from a theoretical point of view. Unexpectedly, 2-qubit systems emerge as a special case outside this scheme. Furthermore, we present a method to reduce the number of CNOT gates, thus deriving a new implementable scaling scheme, which requires only one single layer and whose performance has been tested with a variety of different unitary matrices via the PennyLane library.
In this work, a novel quantum neural network is introduced as a means to approximate any unitary evolution through the Standard Recursive Block Basis (SRBB) and is subsequently redesigned with the number of CNOTs asymptotically reduced by an exponential contribution. This algebraic approach to the problem of unitary synthesis exploits Lie algebras and their topological features to obtain scalable parameterizations of unitary operators. First, the original SRBB-based scalability scheme, already known in the literature only from a theoretical point of view, is reformulated for efficient algorithm implementation and complexity management. Remarkably, 2-qubit operators emerge as a special case of the original scaling scheme. Furthermore, an algorithm is proposed to reduce the number of CNOT gates in the scalable variational quantum circuit, thus deriving a new implementable scaling scheme that requires only one layer of approximation. The single layer CNOT-reduced quantum neural network is implemented, and its performance is assessed with a variety of different unitary matrices, both sparse and dense, up to 6 qubits via the PennyLane library. The effectiveness of the approximation is measured with different metrics in relation to two optimizers: a gradient-based method and the Nelder-Mead method. The approximate CNOT-reduced SRBB-based synthesis algorithm is also tested on real hardware and compared with other valid approximation and decomposition methods available in the literature.
In this paper, we design quantum circuits for the exponential of scaled
n-qubit Pauli strings using single-qubit rotation gates, Hadamard gate, and CNOT gates. A key result we derive is that any two Pauli-string operators composed of identity and
X gates are permutation similar, and the corresponding permutation matrices are product of CNOT gates, with the
n-th qubit serving as the control qubit. Consequently, we demonstrate that the proposed circuit model for exponential of any Pauli-string is implementable on low-connected quantum hardware and scalable i.e. quantum circuits for
(n+1)-qubit systems can be constructed from
n-qubit circuits by adding additional quantum gates and the extra qubit. We then apply these circuit models to approximate unitary evolution for several classes of Hamiltonians using the Suzuki-Trotter approximation. These Hamiltonians include
2-sparse block-diagonal Hamiltonians, Ising Hamiltonians, and both time-independent and time-dependent Random Field Heisenberg Hamiltonians and Transverse Magnetic Random Quantum Ising Hamiltonians. Simulations for systems of up to 18 qubits show that the circuit approximation closely matches the exact evolution, with errors comparable to the numerical Trotterization error. Finally, we consider noise models in quantum circuit simulations to account for gate implementation errors in NISQ computers and observe that the noisy simulation closely resembles the noiseless one when gate and idle errors are on the order of
O(10−3) or smaller.
In optimization, one of the well-known classical algorithms is power iterations. Simply stated, the algorithm recovers the dominant eigenvector of some diagonalizable matrix. Since numerous optimization problems can be formulated as an eigenvalue/eigenvector search, this algorithm features wide applicability. Operationally, power iterations consist of performing repeated matrix-to-vector multiplications (or MatVec) followed by a renormilization step in order to converge to the dominant eigenvalue/eigenvector. However, classical realizations, including novel tensor network based approaches, necessitate an exponential scaling for the algorithm's run-time. In this paper, we propose a quantum realiziation to circumvent this pitfall. Our methodology involves casting low-rank representations; Matrix Product Operators (MPO) for matrices and Matrix Product States (MPS) for vectors, into quantum circuits. Specifically, we recover a unitary approximation by variationally minimizing the Frobenius distance between a target MPO and an MPO ansatz wherein the tensor cores are constrained to unitaries. Such an unitary MPO can easily be implemented as a quantum circuit with the addition of ancillary qubits. Thereafter, with appropriate initialization and post-selection on the ancillary space, we realize a single iteration of the classical algorithm. With our proposed methodology, power iterations can be realized entirely on a quantum computer via repeated, static circuit blocks; therefore, a run-time advantage can indeed be guaranteed. Moreover, by exploiting Riemannian optimization and cross-approximation techniques, our methodology becomes instance agnostic and thus allows one to address black-box optimization within the framework of quantum computing.