Adversary lower bounds in the Hamiltonian oracle model

The adversary method is one of the two main techniques for proving lower bounds in the quantum query model (the other being the polynomial method). It is an extremely versatile method with several equivalent formulations which has been used to obtain good lower bounds for a variety of functions. It can be understood in terms of weight schemes [Amb06,Zha05], via semidefinite programming and spectral analysis [BSS03], or through Kolmogorov complexity [LM03]. All of these formulations have been shown to be equal both in their power and in their limitations [ ŠS06]. Later, an extension of the adversary method was introduced which allows the use of negative weights and removes some of the limitations of the method [HL Š07].

2 Discrete oracles, fractional oracles, and Hamiltonian oracles Suppose that we wish to compute some function f : {0, 1} N → {0, 1}, given the input variables x = x 1 x 2 • • • x N , using a quantum algorithm. The state of the algorithm at any time t, on the input string x, may be written in terms of a set of basis states |j, k such that the first ⌈log N ⌉ qubits j range over the indices of the variables:

In the conventional (discrete) quantum query model, access to the variables is allowed only through a discrete oracle, which can be queried with index j to obtain the value of the variable x j . The query complexity of any particular algorithm computing f is the number of queries made by that algorithm, and the query complexity of the function f itself is the minimum query complexity of any algorithm computing f . In this model, we typically 1 define the query transformation Q x so that the basis state |j, k queries the variable x j , and gains a negative phase if x j = 1. Then the query maps |j, k to (-1) xj |j, k , that is,

In addition to queries, a discrete quantum query algorithm can also perform arbitrary unitary transformations that do not depend on the input string x. An algorithm that makes T discrete queries (T is an integer) is just a sequence of operations alternating between arbitrary unitary transformations and queries:

The sequence is applied to the initial state ψ 0 (which is independent of the input x) to produce the final state ψ T

x , which is measured by the algorithm to produce the output. If the output is correct with probability at least 2 3 , we say that the algorithm computes f with bounded error. The fractional quantum query model generalizes the discrete model by allowing fractions of an oracle query to be made. For integer M , the fractional query Q 1/M x maps |j, k to e -iπ/M xj |j, k . An algorithm in this model is a sequence of operations alternating between arbitrary unitary transformations and such fractional queries:

The Hamiltonian oracle model, introduced in [FG98], results from taking the limit M → ∞ in the fractional query model. It is thus a continuous-time generalization of the discrete query model (see [Moc07,FGG07]). In this model, the state of a quantum algorithm |ψ t

x evolves according to the Schrödinger equation

where H x (t) is the Hamiltonian of the algorithm. The algorithm starts in the initial state ψ 0 and evolves for a time T to reach the final state ψ T x . The query complexity of a function f is then the mininum time T needed to compute f .

The Hamiltonian H x (t) may be decomposed into two parts, a Hamiltonian oracle H Q (x) that depends on the input string x but is independent of time, and a driver Hamiltonian H D (t) that depends on the time t but is independent of the input. (Thus, the Hamiltonian oracle corresponds to the oracles calls and the driver Hamiltonian corresponds to the arbitrary unitary transformations in the discrete query model.) To be as general as possible, we can write the combined Hamiltonian, on the input string x, as

for some |g(t)| ≤ 1.

The Hamiltonian oracle H Q (x) has the form

where each H j operates on an orthogonal subspace V j . That is, writing P j as the projection onto V j , we have H j = P j H j P j . We also assume that ||H j || ≤ 1. For each j, there are two possible operators H (xj ) j , corresponding to x j = 0 and x j = 1.

To simulate the fractional or discrete query model using the Hamiltonian query model, let H j be the matrix with π •x j in the j-th row and column, and zeroes elsewhere. Then each H j operates on an orthogonal subspace. Note that H Q (x) is simply the matrix with the string x on the first N entries of the diagonal, multiplied by π, and zeroes everywhere else. If we now choose g(t) = 1 and H D (t) = 0 and evolve the basis state |j, k for a time 1/M , the result will be the state e -iπ/M xj |j, k , which simulates an oracle call. Likewise, an arbitrary unitary U that is independent of the input may be simulated by setting g(t) = 0 and choosing H D (t) appropriately.

The primary idea behind the adversary method is that if an algorithm computes a function, then it must be able to distinguish between inputs that map to different outputs. A certain amount of information ab

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