State conversion generalizes query complexity to the problem of converting between two input-dependent quantum states by making queries to the input. We characterize the complexity of this problem by introducing a natural information-theoretic norm that extends the Schur product operator norm. The complexity of converting between two systems of states is given by the distance between them, as measured by this norm. In the special case of function evaluation, the norm is closely related to the general adversary bound, a semi-definite program that lower-bounds the number of input queries needed by a quantum algorithm to evaluate a function. We thus obtain that the general adversary bound characterizes the quantum query complexity of any function whatsoever. This generalizes and simplifies the proof of the same result in the case of boolean input and output. Also in the case of function evaluation, we show that our norm satisfies a remarkable composition property, implying that the quantum query complexity of the composition of two functions is at most the product of the query complexities of the functions, up to a constant. Finally, our result implies that discrete and continuous-time query models are equivalent in the bounded-error setting, even for the general state-conversion problem.
We show that any boolean function can be evaluated optimally by a quantum query algorithm that alternates a certain fixed, input-independent reflection with a second reflection that coherently queries the input string. Originally introduced for solving the unstructured search problem, this two-reflections structure is therefore a universal feature of quantum algorithms.Our proof goes via the general adversary bound, a semi-definite program (SDP) that lower-bounds the quantum query complexity of a function. By a quantum algorithm for evaluating span programs, this lower bound is known to be tight up to a sub-logarithmic factor. The extra factor comes from converting a continuous-time query algorithm into a discrete-query algorithm. We give a direct and simplified quantum algorithm based on the dual SDP, with a bounded-error query complexity that matches the general adversary bound.Therefore, the general adversary lower bound is tight; it is in fact an SDP for quantum query complexity. This implies that the quantum query complexity of the composition f o (g,..., g) of two boolean functions f and g matches the product of the query complexities of f and g, without a logarithmic factor for error reduction. It efficiently characterizes the quantum query complexity of a read-once formula over any finite gate set. It further shows that span programs are equivalent to quantum query algorithms.
We show that any boolean function can be evaluated optimally by a quantum query algorithm that alternates a certain fixed, input-independent reflection with a second reflection that coherently queries the input string. Originally introduced for solving the unstructured search problem, this two-reflections structure is therefore a universal feature of quantum algorithms. Our proof goes via the general adversary bound, a semi-definite program (SDP) that lower-bounds the quantum query complexity of a function. By a quantum algorithm for evaluating span programs, this lower bound is known to be tight up to a sub-logarithmic factor. The extra factor comes from converting a continuous-time query algorithm into a discrete-query algorithm. We give a direct and simplified quantum algorithm based on the dual SDP, with a bounded-error query complexity that matches the general adversary bound. Therefore, the general adversary lower bound is tight; it is in fact an SDP for quantum query complexity. This implies that the quantum query complexity of the composition f ∘ (g, …, g) of two boolean functions f and g matches the product of the query complexities of f and g, without a logarithmic factor for error reduction. It efficiently characterizes the quantum query complexity of a read-once formula over any finite gate set. It further shows that span programs are equivalent to quantum query algorithms.
Quantum query complexity measures the number of input bits that must be read by a quantum algorithm in order to evaluate a function. Hoyer et al. (2007) have generalized the adversary semidefinite program that lower-bounds quantum query complexity. By giving a matching quantum algorithm, we show that the general adversary lower bound is tight for every boolean function. The proof is based on span programs, a linear-algebraic computational model without inherent dynamics. Span programs correspond to solutions to the dual semi-definite program, and to bipartite graphs. The analysis shows that properties of eigenvalue-zero eigenvectors of the graphs in fact imply an “effective” spectral gap around zero We thus develop a quantum algorithm for evaluating span programs. It follows that span programs, measured by witness size, and quantum algorithms, measured by query complexity, are equivalent computational models, up to a constant factor. The result efficiently characterizes the quantum query complexity of a read-once formula over any finite gate set. It also implies that the quantum query complexity of the composition f ◦ (g, . . . , g) of two boolean functions matches the product of the query complexities of f and g, without a logarithmic factor for error reduction. The algorithm alternates a fixed reflection with input queries. Originally introduced for solving the unstructured search problem, this structure is therefore a universal feature of quantum query algorithms. We give a second algorithm for evaluating span programs that has the further potential to be time-efficient. Subsequent applications have derived nearly time-optimal quantum algorithms for evaluating many read-once formulas. Span programs may have promise for developing more quantum algorithms.
Quantum query complexity enjoys a variety of pleasing joint computation properties: for example, a composition theorem asserting
Q(f∘g)=Θ(Q(f)Q(g)) for all Boolean functions
f and
g; a direct sum theorem asserting that computing
k copies of a function (or search problem) costs
Ω(k) times as much as the cost of computing one copy; and a direct product theorem asserting that for Boolean functions, even succeeding at the direct sum problem with exponentially small probability still requires
Ω(k) times the cost of computing one copy to bounded error.
However, all of these results are strictly for worst-case quantum query complexity. For example, if we have a fixed distribution
μ over inputs, the direct sum theorem says nothing about the quantum query complexity of computing
k copies of
f when the input comes from the product distribution
μk instead of being worst-case. (Note that while a standard Yao-type minimax theorem guarantees a hard distribution for the direct sum problem, there's no guarantee that this hard distribution is a product distribution.) A similar problem occurs for the direct product theorem and the composition theorem: none of these results respect distributions.
In this work, we give distributional joint computation lower bounds for the composition, direct sum, and direct product problems. Along the way, we introduce some new tools for handling quantum query lower bounds, including (a) a new ``multiplicative'' variant of the
γ2 norm (which we use in place of the multiplicative adversary method for proving the direct product theorem), and (b) a new ``Shaltiel-free'' measure of quantum query complexity, which we show characterizes the composition behavior of distributional quantum query complexity and satisfies pleasing properties.
We show that any boolean function can be evaluated optimally by a quantum query algorithm that alternates a certain fixed, input-independent reflection with a second reflection that coherently queries the input string. Originally introduced for solving the unstructured search problem, this two-reflections structure is therefore a universal feature of quantum algorithms. Our proof goes via the general adversary bound, a semi-definite program (SDP) that lower-bounds the quantum query complexity of a function. By a quantum algorithm for evaluating span programs, this lower bound is known to be tight up to a sub-logarithmic factor. The extra factor comes from converting a continuous-time query algorithm into a discrete-query algorithm. We give a direct and simplified quantum algorithm based on the dual SDP, with a bounded-error query complexity that matches the general adversary bound. Therefore, the general adversary lower bound is tight; it is in fact an SDP for quantum query complexity. This implies that the quantum query complexity of the composition f(g,...,g) of two boolean functions f and g matches the product of the query complexities of f and g, without a logarithmic factor for error reduction. It further shows that span programs are equivalent to quantum query algorithms.
The general adversary bound is a semi-definite program that lower-bounds the number of input queries needed by a quantum algorithm to evaluate a function. It is known to be tight up to constant factors for functions (total or partial) with boolean output and binary input alphabet. We show that the general adversary bound is tight for any function whatsoever, with potentially non-boolean input or output alphabets. We also show that quantum query complexity exhibits a remarkable composition property: Q ( f(g(x), . . . , g(x)) ) = O ( Q(f)Q(g) ) for any compatible functions f, g. This was previously known only in the boolean case. Both of these results are obtained by defining a new, but closely related, semi-definite program that we call the witness size. The minimization formulation of the witness size adds constraints compared with the general adversary bound to enable dual solutions to correspond to eigenvalue-zero eigenvectors of certain graphs. While the witness size can be strictly larger than the adversary bound, we show that it can be at most a factor of two larger.