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.
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.
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.
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 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 $\sparseness$ (the maximum number of nonzero elements in a column) and the evolution time t, whereas previous methods had complexity scaling as $\sparseness^4$ 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 $\UN\times\UN$ unitary operation use $\tilde O(\UN^2)$ elementary gates, we show that a black-box unitary can be performed with bounded error using $O(\UN^{2/3} (\log\log \UN)^{4/3})$ queries to its matrix elements. In fact, except for pathological cases, it appears that most unitaries can be performed with only $\tilde O(\sqrt{\UN})$ 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 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.
The problem considered in this thesis is the following: We are given a Hamiltonian H and time t, and our goal is to approximately implement the unitary operator e^{-iHt} with an efficient quantum algorithm. We present an efficient algorithm for simulating sparse Hamiltonians. In terms of the maximum degree d and dimension N of the space on which the Hamiltonian acts, this algorithm uses (d^2(d+log^* N)||Ht||)^{1+o(1)} queries. This improves the complexity of the sparse Hamiltonian simulation algorithm of Berry, Ahokas, Cleve, and Sanders, which scales like (d^4(log^* N)||Ht||)^{1+o(1)}. In terms of the parameter t, these algorithms are essentially optimal due to a no--fast-forwarding theorem. \n \nIn the second part of this thesis, we consider non-sparse Hamiltonians and show significant limitations on their simulation. We generalize the no--fast-forwarding theorem to dense Hamiltonians, and rule out generic simulations taking time o(||Ht||), even though ||H|| is not a unique measure of the size of a dense Hamiltonian H. We also present a stronger limitation ruling out the possibility of generic simulations taking time poly(||Ht||,log N), showing that known simulations based on discrete-time quantum walks cannot be dramatically improved in general. We also show some positive results about simulating structured Hamiltonians efficiently.