In computer science, many search problems are reducible to decision problems, which implies that finding a solution is as hard as deciding whether a solution exists. A quantum analogue of search-to-decision reductions would be to ask whether a quantum algorithm with access to a $\mathsf{QMA}$ oracle can construct $\mathsf{QMA}$ witnesses as quantum states. By a result from Irani, Natarajan, Nirkhe, Rao, and Yuen (CCC '22), it is known that this does not hold relative to a quantum oracle, unlike the cases of $\mathsf{NP}$ , $\mathsf{MA}$ , and $\mathsf{QCMA}$ where search-to-decision relativizes. We prove that if one is not interested in the quantum witness as a quantum state but only in terms of its partial assignments, i.e. the reduced density matrices, then there exists a classical polynomial-time algorithm with access to a $\mathsf{QMA}$ oracle that outputs approximations of the density matrices of a near-optimal quantum witness, for any desired constant locality and inverse polynomial error. Our construction is based on a circuit-to-Hamiltonian mapping that approximately preserves near-optimal $\mathsf{QMA}$ witnesses and a new $\mathsf{QMA}$ -complete problem, Low-energy Density Matrix Verification, which is called by the $\mathsf{QMA}$ oracle to adaptively construct approximately consistent density matrices of a low-energy state.In computer science, many search problems are reducible to decision problems, which implies that finding a solution is as hard as deciding whether a solution exists. A quantum analogue of search-to-decision reductions would be to ask whether a quantum algorithm with access to a $\mathsf{QMA}$ oracle can construct $\mathsf{QMA}$ witnesses as quantum states. By a result from Irani, Natarajan, Nirkhe, Rao, and Yuen (CCC '22), it is known that this does not hold relative to a quantum oracle, unlike the cases of $\mathsf{NP}$, $\mathsf{MA}$, and $\mathsf{QCMA}$ where search-to-decision relativizes. We prove that if one is not interested in the quantum witness as a quantum state but only in terms of its partial assignments, i.e. the reduced density matrices, then there exists a classical polynomial-time algorithm with access to a $\mathsf{QMA}$ oracle that outputs approximations of the density matrices of a near-optimal quantum witness, for any desired constant locality and inverse polynomial error. Our construction is based on a circuit-to-Hamiltonian mapping that approximately preserves near-optimal $\mathsf{QMA}$ witnesses and a new $\mathsf{QMA}$-complete problem, Low-energy Density Matrix Verification, which is called by the $\mathsf{QMA}$ oracle to adaptively construct approximately consistent density matrices of a low-energy state.

arXiv:2408.03986Aug 7, 20240 citations

Tridiagonal matrix decomposition for Hamiltonian simulation on a quantum computer

Boris Arseniev, Dmitry Guskov, Richik Sengupta +2 more

The construction of quantum circuits to simulate Hamiltonian evolution is central to many quantum algorithms. State-of-the-art circuits are based on oracles whose implementation is often omitted, and the complexity of the algorithm is estimated by counting oracle queries. However, in practical applications, an oracle implementation contributes a large constant factor to the overall complexity of the algorithm. The key finding of this work is the efficient procedure for representation of a tridiagonal matrix in the Pauli basis, which allows one to construct a Hamiltonian evolution circuit without the use of oracles. The procedure represents a general tridiagonal matrix ${2}^{n}\ifmmode\times\else\texttimes\fi{}{2}^{n}$ by systematically determining all Pauli strings present in the decomposition, dividing them into commuting subsets. The efficiency is in the number of commuting subsets $O(n)$ . The method is demonstrated using the one-dimensional wave equation, verifying numerically that the gate complexity as a function of the number of qubits is lower than the oracle-based approach for $n<15$ and requires half the number of qubits. This method is applicable to other Hamiltonians based on the tridiagonal matrices.The construction of quantum circuits to simulate Hamiltonian evolution is central to many quantum algorithms. State-of-the-art circuits are based on oracles whose implementation is often omitted, and the complexity of the algorithm is estimated by counting oracle queries. However, in practical applications, an oracle implementation contributes a large constant factor to the overall complexity of the algorithm. The key finding of this work is the efficient procedure for representation of a tridiagonal matrix in the Pauli basis, which allows one to construct a Hamiltonian evolution circuit without the use of oracles. The procedure represents a general tridiagonal matrix ${2}^{n}\ifmmode\times\else\texttimes\fi{}{2}^{n}$ by systematically determining all Pauli strings present in the decomposition, dividing them into commuting subsets. The efficiency is in the number of commuting subsets $O(n)$. The method is demonstrated using the one-dimensional wave equation, verifying numerically that the gate complexity as a function of the number of qubits is lower than the oracle-based approach for $n<15$ and requires half the number of qubits. This method is applicable to other Hamiltonians based on the tridiagonal matrices.

arXiv:2310.00121Physical Review AMay 29, 20244 citations

Quantum logic gate construction and quantum algorithms

Xinlan Zhou

Jan 1, 20020 citations

Quantum logic gates and circuits construction

Piotr Marecki

Archiwum Informatyki Teoretycznej i StosowanejJan 1, 20010 citations

\mathsf{QMA}

oracle can construct

\mathsf{QMA}

witnesses as quantum states. By a result from Irani, Natarajan, Nirkhe, Rao, and Yuen (CCC '22), it is known that this does not hold relative to a quantum oracle, unlike the cases of

\mathsf{NP}

\mathsf{MA}

, and

\mathsf{QCMA}

where search-to-decision relativizes. We prove that if one is not interested in the quantum witness as a quantum state but only in terms of its partial assignments, i.e. the reduced density matrices, then there exists a classical polynomial-time algorithm with access to a

\mathsf{QMA}

oracle that outputs approximations of the density matrices of a near-optimal quantum witness, for any desired constant locality and inverse polynomial error. Our construction is based on a circuit-to-Hamiltonian mapping that approximately preserves near-optimal

\mathsf{QMA}

witnesses and a new

\mathsf{QMA}

-complete problem, Low-energy Density Matrix Verification, which is called by the

\mathsf{QMA}

oracle to adaptively construct approximately consistent density matrices of a low-energy state.

O(n)

. The method is demonstrated using the one-dimensional wave equation, verifying numerically that the gate complexity as a function of the number of qubits is lower than the oracle-based approach for

n<15

and requires half the number of qubits. This method is applicable to other Hamiltonians based on the tridiagonal matrices.