We present a quantum algorithm for approximating the real time evolution e−iHt of an arbitrary d-sparse Hamiltonian to error є, given black-box access to the positions and b-bit values of its non-zero matrix entries. The query complexity of our algorithm is O((t√d||H||1 → 2)1+o(1)/єo(1)) with respect to the largest Euclidean row norm ||H||1 → 2, which is shown to be optimal up to subpolynomial factors through a matching lower bound, and it uses a factor O(b) more gates. This provides a polynomial speedup in sparsity for the common case where the spectral norm is known, and generalizes previous approaches which achieve optimal scaling, but with respect to more restrictive parameters. By exploiting knowledge of the spectral norm, our algorithm solves the black-box unitary implementation problem – O(d1/2+o(1)) queries suffice to approximate any d-sparse unitary in the black-box setting, which matches the quantum search lower bound of Ω(√d) queries and improves upon prior art [Berry and Childs, QIP 2010] of Õ(d2/3) queries. Combined with known techniques, we also solve systems of sparse linear equations with condition number κ using O((κ √d)1+o(1)/єo(1)) queries, which is a quadratic improvement in sparsity.
We present a quantum algorithm for approximating the real time evolution e−iHt of an arbitrary d-sparse Hamiltonian to error є, given black-box access to the positions and b-bit values of its non-zero matrix entries. The query complexity of our algorithm is O((t√d||H||1 → 2)1+o(1)/єo(1)) with respect to the largest Euclidean row norm ||H||1 → 2, which is shown to be optimal up to subpolynomial factors through a matching lower bound, and it uses a factor O(b) more gates. This provides a polynomial speedup in sparsity for the common case where the spectral norm is known, and generalizes previous approaches which achieve optimal scaling, but with respect to more restrictive parameters. By exploiting knowledge of the spectral norm, our algorithm solves the black-box unitary implementation problem – O(d1/2+o(1)) queries suffice to approximate any d-sparse unitary in the black-box setting, which matches the quantum search lower bound of Ω(√d) queries and improves upon prior art [Berry and Childs, QIP 2010] of Õ(d2/3) queries. Combined with known techniques, we also solve systems of sparse linear equations with condition number κ using O((κ √d)1+o(1)/єo(1)) queries, which is a quadratic improvement in sparsity.
We give a quantum algorithm for simulating a d-sparse Hermitian Hamiltonian H, assuming a known upper bound Λ on its maximum column Euclidean norm ∥H∥1→2. For tΛ≥1/2, simulation with operator-norm error ε uses O(tΛd+dlog(2/ε)) sparse-oracle queries. This removes the subpolynomial overhead in Low's algorithm [STOC 2019], replacing it with an additive logarithmic precision term. For d>1 and tΛ≥log(2/ε), the bound matches the worst-case lower bound. A known spectral-norm upper bound may also be used in place of Λ. The number of 1- and 2-qubit gates is linear in the query scale, up to oracle costs and polynomial overhead in the input bit lengths and logarithmic precision parameters.
As applications, we obtain O(κdpolylog(κ/ε)) queries for solving d-sparse quantum linear systems with ∥A∥≤1 and ∥A−1∥≤κ, under standard sparse and state-preparation access. We also give a gate-efficient implementation of black-box unitaries with at most d nonzero entries per row and column using O(dlog(2/ε)) queries, given sparse access to the unitary and its adjoint. At constant error, the query bound is optimal and yields Θ(N) queries for arbitrary N×N unitaries, resolving the open question on black-box unitary implementation posed by Berry and Childs [QIC 2012].
We give quantum algorithms for Hamiltonian simulation, linear differential equations, and linear systems with square-root dependence on sparsity. For an s-sparse Hermitian matrix with row and column Euclidean norms bounded by η, evolution for time t to error ε uses O(sτ+sτlog(1/ε)+log(1/ε)) queries, where τ=η∣t∣. The same bound holds for time-dependent Hamiltonians with coherent time-indexed sparse access and certified time discretisation. Our construction combines an exact entrywise representation with Cayley transducers. A joint lower bound nearly matches the time, sparsity, and precision dependence, ruling out a uniformly additive bound in sτ and log(1/ε). For dissipative linear differential equations we obtain O(gˉ[s(τ+τ)+1]log(Cgˉ/ε)) queries, where τ is the evolution time multiplied by a norm bound, gˉ bounds the ratio of combined input and source norms to the final-solution norm, and C is an absolute constant. We sharpen this bound by separating Hamiltonian and dissipative components, and treat general constant generators. Lower bounds capture joint parameter dependence for time-dependent equations and precision hardness for constant positive semidefinite dissipation. For linear systems, a Cayley walk and Chebyshev filter give O(κslog(1/ε)) queries when ∥A∥≤1 and ∥A−1∥≤κ, and we give an alternative proof of a matching joint lower bound. We also give gate-efficient implementations of time-independent Hamiltonian simulation and the dissipative ODE solver, including Lipschitz time dependence, retaining the query bounds with polylogarithmic overhead in inverse precision given efficient entry decoding, arithmetic, and required source preparations.
We present general methods for simulating black-box Hamiltonians using quantum walks. These techniques have two main applications: simulating sparse Hamiltonians and implementing black-box unitary operations. In particular, we give the best known simulation of sparse Hamiltonians with constant precision. Our method has complexity linear in both the sparseness D (the maximum number of nonzero elements in a column) and the evolution time t, whereas previous methods had complexity scaling as D^4 and were superlinear in t. We also consider the task of implementing an arbitrary unitary operation given a black-box description of its matrix elements. Whereas standard methods for performing an explicitly specified N x N unitary operation use O(N^2) elementary gates, we show that a black-box unitary can be performed with bounded error using O(N^{2/3} (log log N)^{4/3}) queries to its matrix elements. In fact, except for pathological cases, it appears that most unitaries can be performed with only O(sqrt{N}) queries, which is optimal.
In the quantum linear systems problem (QLSP), we are given query access to a d-sparse N×N matrix A with condition number κ, and the ability to prepare a quantum state proportional to a vector b. The goal is to output an ε-approximation to the quantum state proportional to the solution x of Ax=b. Following a long line of work, the best previously known quantum algorithms for the QLSP had query complexities O(κdlog(1/ε)) and κd(κd/ε)o(1), while the best known lower bounds were Ω(κlog(1/ε)) and Ω(κd). We improve these bounds and show that the complexity of the QLSP is Θ(κdlog(1/ε)). We also resolve an open problem of Berry and Childs by showing that any N×N unitary can be implemented with bounded error using O(N) queries to its matrix entries.
We present the problem of approximating the time-evolution operator e-iHt to error ϵ, where the Hamiltonian H = ((G|⊗I)U (|Gi⊗I) is the projection of a unitary oracle U onto the state |Gi created by another unitary oracle. Our algorithm solves this with a query complexity O ( t + log(1/ϵ) ) to both oracles that is optimal with respect to all parameters in both the asymptotic and non-asymptotic regime, and also with low overhead, using at most two additional ancilla qubits. This approach to Hamiltonian simulation subsumes important prior art considering Hamiltonians which are d-sparse or a linear combination of unitaries, leading to significant improvements in space and gate complexity, such as a quadratic speed-up for precision simulations. It also motivates useful new instances, such as where H is a density matrix. A key technical result is 'qubitization', which uses the controlled version of these oracles to embed any H in an invariant SU(2) subspace. A large class of operator functions of H can then be computed with optimal query complexity, of which e-iHt is a special case.
We present general methods for simulating black-box Hamiltonians using quantum walks. These techniques have two main applications: simulating sparse Hamiltonians and implementing black-box unitary operations. In particular, we give the best known simulation of sparse Hamiltonians with constant precision. Our method has complexity linear in both the sparseness D (the maximum number of nonzero elements in a column) and the evolution time t, whereas previous methods had complexity scaling as D4 and were superlinear in t. We also consider the task of implementing an arbitrary unitary operation given a black-box description of its matrix elements. Whereas standard methods for performing an explicitly specified N × N unitary operation use O(N2) elementary gates, we show that a black-box unitary can be performed with bounded error using O(N2/3(log logN)4/3) queries to its matrix elements. In fact, except for pathological cases, it appears that most unitaries can be performed with only O(√N) queries, which is optimal.