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تسجيل مستخدم جديدQuantum Circuits with Unbounded Fan-out
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Peter Hoyer
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We demonstrate that the unbounded fan-out gate is very powerful. Constant-depth polynomial-size quantum circuits with bounded fan-in and unbounded fan-out over a fixed basis (denoted by QNCf^0) can approximate with polynomially small error the following gates: parity, mod[q], And, Or, majority, threshold[t], exact[q], and Counting. Classically, we need logarithmic depth even if we can use unbounded fan-in gates. If we allow arbitrary one-qubit gates instead of a fixed basis, then these circuits can also be made exact in log-star depth. Sorting, arithmetical operations, phase estimation, and the quantum Fourier transform with arbitrary moduli can also be approximated in constant depth.
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Recently, Bravyi, Gosset, and K{o}nig (Science, 2018) exhibited a search problem called the 2D Hidden Linear Function (2D HLF) problem that can be solved exactly by a constant-depth quantum circuit using bounded fan-in gates (or QNC^0 circuits), but
cannot be solved by any constant-depth classical circuit using bounded fan-in AND, OR, and NOT gates (or NC^0 circuits). In other words, they exhibited a search problem in QNC^0 that is not in NC^0. We strengthen their result by proving that the 2D HLF problem is not contained in AC^0, the class of classical, polynomial-size, constant-depth circuits over the gate set of unbounded fan-in AND and OR gates, and NOT gates. We also supplement this worst-case lower bound with an average-case result: There exists a simple distribution under which any AC^0 circuit (even of nearly exponential size) has exponentially small correlation with the 2D HLF problem. Our results are shown by constructing a new problem in QNC^0, which we call the Relaxed Parity Halving Problem, which is easier to work with. We prove our AC^0 lower bounds for this problem, and then show that it reduces to the 2D HLF problem. As a step towards even stronger lower bounds, we present a search problem that we call the Parity Bending Problem, which is in QNC^0/qpoly (QNC^0 circuits that are allowed to start with a quantum state of their choice that is independent of the input), but is not even in AC^0[2] (the class AC^0 with unbounded fan-in XOR gates). All the quantum circuits in our paper are simple, and the main difficulty lies in proving the classical lower bounds. For this we employ a host of techniques, including a refinement of H{aa}stads switching lemmas for multi-output circuits that may be of independent interest, the Razborov-Smolensky AC^0[2] lower bound, Vaziranis XOR lemma, and lower bounds for non-local games.
Instruction scheduling is a key compiler optimization in quantum computing, just as it is for classical computing. Current schedulers optimize for data parallelism by allowing simultaneous execution of instructions, as long as their qubits do not ove
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