. 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.

DOI: 10.26421/qic12.1-2-4

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Aug 18, 20102 citations

\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.

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\to2}

. For

tΛ\ge1/2

, simulation with operator-norm error

ε

uses

O\!\left(tΛ\sqrt d+\sqrt d\log(2/ε)\right)

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\ge\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(κ\sqrt d\,\mathrm{polylog}(κ/ε))

queries for solving

d

-sparse quantum linear systems with

\|A\|\le1

and

\|A^{-1}\|\leκ

, 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(\sqrt d\log(2/ε))

queries, given sparse access to the unitary and its adjoint. At constant error, the query bound is optimal and yields

Θ(\sqrt N)

queries for arbitrary

N\times 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

\mathcal{O}(\sqrt{s}τ+\sqrt{sτ\log(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

\sqrt{s}τ

and

\log(1/ε)

. For dissipative linear differential equations we obtain

\mathcal{O}(\bar g[\sqrt{s}(τ+\sqrtτ)+1]\log(C\bar g/ε))

queries, where

τ

is the evolution time multiplied by a norm bound,

\bar g

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

\mathcal{O}(κ\sqrt{s}\log(1/ε))

queries when

\|A\|\le1

and

\|A^{-1}\|\leκ

, 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.

O(N^2)

complexity for dense graphs with

N

nodes. For sparse graphs, the computational cost scales linearly with the number of edges, which is

O(N)

in many cases. We implement the framework in the Python package SQWLib and illustrate its capabilities through simulations of representative algorithms, including quantum simulated annealing and quantum search on graphs. These results provide a practical tool for studying Szegedy-walk-based algorithms numerically beyond purely analytical treatments.