Does Rydberg state manipulation equal quantum computation?

Manipulation Equal Quantum Computation? Ahn et al. (1), in describing the experimental search of a database realized with Rydberg atoms, left the misleading impression that the search constituted quantum computation. Their atomic system, in the words of the accompanying Perspective, “can implement [quantum] algorithms” (2) such as the database-search algorithm proposed by Grover (3, 4). And, according to Ahn et al. [p. 465 of (1)], “[o]ther algorithms can be implemented by other unitary transformations such as the application of ultrafast shaped terahertz pulses” as described by Tielking and Jones (5). The system described by Ahn et al., however, cannot implement quantum algorithms any more efficiently than can a classical digital computer. The word “efficiently” in this context refers to how the resources required to implement an algorithm scale with the size of the problem being solved (6). The standard model for quantum computation uses poly-local transformations (7)—implemented, for example, by polynomially many bounded size gates acting on n qubits. These require specification of only polynomially many nontrivial transition amplitudes with constant precision. Any quantum algorithm within this model can also be implemented (on a classical digital computer, for example) by storing the vector of N 5 2 complex amplitudes, which represents the quantum state at each time step, and by successively multiplying the state vector by the N-by-N matrix that specifies each unitary transformation. The resources required for such an implementation scale exponentially, rather than polynomially, with n; the classical implementation is therefore less efficient. The quantum phase manipulation of Rydberg atom states described by Ahn et al. (1) is not quantum computation because, like this classical implementation, it scales badly with the size of the problem—the number of bits, n 5 log N, defining the size of the quantum register being searched. This scaling is somewhat subtle because it has two causes, one of which Ahn et al. partly acknowledged and the other of which they did not discuss. First, the number of atoms they used for each shot at the single-query Grover algorithm (4) was Nmin ' (2ε) ' 100. To achieve acceptable success rates, Ahn et al. found that increasing N requires increasing numbers of shots [figure 3 of (1)] or, equivalently, more atoms. In fact, because ε 5 O(1/=N), Grover showed that the number of N-state quantum systems required scales at least as N log N (4); this single-query algorithm is thus no more efficient than a classical algorithm for the same problem, which is why only Grover’s O(=N) query algorithm (3) is the one usually discussed in the quantum computing literature (8). Second, there is also an exponential price for removing entanglement from Grover’s algorithms (9, 10), which commonly is paid with exponentially increasing mass/energy (11). Using N Rydberg states rather than n qubits realizes Lloyd’s scheme for removing entanglement (9), but Ahn et al. (1) neglected the exponential overhead required for measurement and for realization of N-by-N unitary transformations. Because the difference (detuning) between adjacent Rydberg energy levels converges to zero polynomially in 1/N (5), both the laser pulses and the final measurements must be specified with exponentially increasing precision in n. To the best of our knowledge, all physically realized computation is quantum mechanical—but that does not make every computer a quantum computer. A quantum system such as a Rydberg atom that implements a quantum algorithm no more efficiently than a classical computer is not performing quantum computation.