Quantum singular value transformation

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Quantum singular value transformation (QSVT) is a framework for building quantum algorithms that rely on linear algebra. It covers many tasks such as simulating quantum systems, searching, and solving linear equations.

QSVT began in 2018, introduced by András Gilyén, Yuan Su, Guang Hao Low, and Nathan Wiebe. It extends earlier Hamiltonian simulation methods that drew ideas from signal processing.

The core idea is block-encoding. A quantum circuit is a block-encoding of a matrix A if it is a larger unitary that contains A in a corner, so that applying the unitary to a simple state recovers A acting on a target state.

The central algorithm of QSVT takes a block-encoding of A and produces a block-encoding of p(A, A†), where p is a polynomial of degree d. This uses only d calls to the block-encoding circuit and a single ancilla qubit. By choosing the polynomial p, this effectively applies a polynomial to the singular values of A, which is why it is called a singular value transformation.

There is a Hermitian special case: if A is Hermitian, the method can perform an eigenvalue transformation. If A has eigen-decomposition A = sum_i lambda_i |u_i>
Together, these ideas form a flexible toolkit for many quantum algorithms that depend on manipulating spectral properties of matrices.