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Quantum fanout is powerful
 Theory of Computing
"... We demonstrate that the unbounded fanout gate is very powerful. Constantdepth polynomialsize quantum circuits with bounded fanin and unbounded fanout over a fixed basis (denoted by QNC 0 f) can approximate with polynomially small error the following gates: parity, mod[q], And, Or, majority, thr ..."
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We demonstrate that the unbounded fanout gate is very powerful. Constantdepth polynomialsize quantum circuits with bounded fanin and unbounded fanout over a fixed basis (denoted by QNC 0 f) can approximate with polynomially small error the following gates: parity, mod[q], And, Or, majority, threshold[t], exact[t], and Counting. Classically, we need logarithmic depth even if we can use unbounded fanin gates. If we allow arbitrary onequbit gates instead of a fixed basis, then these circuits can also be made exact in logstar depth. Sorting, arithmetic operations, phase estimation, and the quantum Fourier transform with arbitrary moduli can also be approximated in constant depth. 1
Universal Quantum Circuits
"... Abstract. We define and construct efficient depthuniversal and almostsizeuniversal quantum circuits. Such circuits can be viewed as generalpurpose simulators for central classes of quantum circuits and can be used to capture the computational power of the circuit class being simulated. For depth w ..."
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Abstract. We define and construct efficient depthuniversal and almostsizeuniversal quantum circuits. Such circuits can be viewed as generalpurpose simulators for central classes of quantum circuits and can be used to capture the computational power of the circuit class being simulated. For depth we construct universal circuits whose depth is the same order as the circuits being simulated. For size, there is a log factor blowup in the universal circuits constructed here. We prove that this construction is nearly optimal. 1
Simulating Special but Natural Quantum Circuits
"... We identify a subclass of BQP that captures certain structural commonalities among many quantum algorithms including Shor’s algorithms. This class does not contain all of BQP (e.g. Grover’s algorithm does not fall into this class). Our main result is that any algorithm in this class that measures a ..."
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We identify a subclass of BQP that captures certain structural commonalities among many quantum algorithms including Shor’s algorithms. This class does not contain all of BQP (e.g. Grover’s algorithm does not fall into this class). Our main result is that any algorithm in this class that measures at most O(log n) qubits can be simulated by classical randomized polynomial time algorithms. This does not dequantize Shor’s algorithm (as the latter measures n qubits) but our work also highlights a new potentially hard function for cryptographic applications. Our main technical contribution is (to the best of our knowledge) a new exact characterization of certain sums of Fouriertype coefficients (with exponentially many summands). One of the key problems in complexity theory is to determine the power of the complexity class BQP. Recall that this is the set of languages accepted by uniform polynomial size quantum circuits with bounded twosided error. It is essentially the quantum version of the complexity class BPP. Just as BPP corresponds to what is feasible on a classical computer with randomness, BQP corresponds
Simple Proof of Polylog Counting Ability of FirstOrder Logic. ” And wishing everyone a theoremfilled
"... Warmest thanks to Debajyoti, Fred, and Steve for this issue’s column on smalldepth quantum circuits. Please stay tuned to future columns to hear from Salil Vadhan on the complexity of zero knowledge and from Arnaud Durand, (the late) Clemens Lautemann, and Malika More on “A ..."
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Warmest thanks to Debajyoti, Fred, and Steve for this issue’s column on smalldepth quantum circuits. Please stay tuned to future columns to hear from Salil Vadhan on the complexity of zero knowledge and from Arnaud Durand, (the late) Clemens Lautemann, and Malika More on “A
A New Lower Bound Technique for Quantum Circuits without Ancillæ
"... We present a technique to derive depth lower bounds for quantum circuits. The technique is based on the observation that in circuits without ancillæ, only a few input states can set all the control qubits of a Toffoli gate to 1. This can be used to selectively remove large Toffoli gates from a quant ..."
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We present a technique to derive depth lower bounds for quantum circuits. The technique is based on the observation that in circuits without ancillæ, only a few input states can set all the control qubits of a Toffoli gate to 1. This can be used to selectively remove large Toffoli gates from a quantum circuit while keeping the cumulative error low. We use the technique to give another proof that parity cannot be computed by constant depth quantum circuits without ancillæ. 1
Quantum Fanout is Powerful
, 2004
"... Abstract: We demonstrate that the unbounded fanout gate is very powerful. Constantdepth polynomialsize quantum circuits with bounded fanin and unbounded fanout over a fixed basis (denoted by QNC0 f) can approximate with polynomially small error the following gates: parity, mod[q], And, Or, major ..."
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Abstract: We demonstrate that the unbounded fanout gate is very powerful. Constantdepth polynomialsize quantum circuits with bounded fanin and unbounded fanout over a fixed basis (denoted by QNC0 f) can approximate with polynomially small error the following gates: parity, mod[q], And, Or, majority, threshold[t], exact[t], and Counting. Classically, we need logarithmic depth even if we can use unbounded fanin gates. If we allow arbitrary onequbit gates instead of a fixed basis, then these circuits can also be made exact in logstar depth. Sorting, arithmetic operations, phase estimation, and the quantum Fourier transform with arbitrary moduli can also be approximated in constant depth. ACM Classification: F.2.1, F.2.2 AMS Classification: 68Q15, 81P68 Key words and phrases: quantum computing, quantum circuits, fanout, quantum Fourier transform, constant depth circuits, threshold circuits 1
Implementing fanout, parity, and Mod gates via spin exchange interactions
, 2004
"... We show that, for any n> 0, the Heisenberg interaction among 2n qubits (as spin1/2 particles) can be used to exactly implement an nqubit parity gate, which is equivalent in constant depth to an nqubit fanout gate. Either isotropic or nonisotropic versions of the interaction can be used. We gen ..."
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We show that, for any n> 0, the Heisenberg interaction among 2n qubits (as spin1/2 particles) can be used to exactly implement an nqubit parity gate, which is equivalent in constant depth to an nqubit fanout gate. Either isotropic or nonisotropic versions of the interaction can be used. We generalize our basic results by showing that any Hamiltonian (acting on suitably encoded logical qubits), whose eigenvalues depend quadratically on the Hamming weight of the logical qubit values, can be used to implement generalized Modq gates for any q ≥ 2. This paper is a sequel to quantph/0309163, and resolves a question left open in that paper. 1