On the Complexity of Quantum ACC

For any $q > 1$, let \MOD_q be a quantum gate that determines if the number of 1's in the input is divisible by $q$. We show that for any $q,t > 1$, \MOD_q is equivalent to \MOD_t (up to constant depth). Based on the case $q=2$, Moore \cite{moore99} has shown that quantum analogs of AC$^{(0)}$, ACC$[q]$, and ACC, denoted QAC$^{(0)}_{wf}$, QACC$[2]$, QACC respectively, define the same class of operators, leaving $q > 2$ as an open question. Our result resolves this question, proving that QAC$^{(0)}_{wf} =$ QACC$[q] =$ QACC for all $q$. We also develop techniques for proving upper bounds for QACC in terms of related language classes. We define classes of languages EQACC, NQACC and BQACC_{\rats}. We define a notion $\log$-planar QACC operators and show the appropriately restricted versions of EQACC and NQACC are contained in P/poly. We also define a notion of $\log$-gate restricted QACC operators and show the appropriately restricted versions of EQACC and NQACC are contained in TC$^{(0)}$. To do this last proof, we show that TC$^{(0)}$ can perform iterated addition and multiplication in certain field extensions. We also introduce the notion of a polynomial-size tensor graph and show that families of such graphs can encode the amplitudes resulting from apply an arbitrary QACC operator to an initial state.Comment: 22 pages, 4 figures This version will appear in the July 2000 Computational Complexity conference. Section 4 has been significantly revised and many typos correcte

Bounds on the Power of Constant-Depth Quantum Circuits

We show that if a language is recognized within certain error bounds by constant-depth quantum circuits over a finite family of gates, then it is computable in (classical) polynomial time. In particular, our results imply EQNC^0 is contained in P, where EQNC^0 is the constant-depth analog of the class EQP. On the other hand, we adapt and extend ideas of Terhal and DiVincenzo (quant-ph/0205133) to show that, for any family F of quantum gates including Hadamard and CNOT gates, computing the acceptance probabilities of depth-five circuits over F is just as hard as computing these probabilities for circuits over F. In particular, this implies that NQNC^0 = NQACC = NQP = coC=P where NQNC^0 is the constant-depth analog of the class NQP. This essentially refutes a conjecture of Green et al. that NQACC is contained in TC^0 (quant-ph/0106017)

Implementing a Fast Unbounded Quantum Fanout Gate Using Power-Law Interactions

The standard circuit model for quantum computation presumes the ability to directly perform gates between arbitrary pairs of qubits, which is unlikely to be practical for large-scale experiments. Power-law interactions with strength decaying as $1/r^\alpha$ in the distance $r$ provide an experimentally realizable resource for information processing, whilst still retaining long-range connectivity. We leverage the power of these interactions to implement a fast quantum fanout gate with an arbitrary number of targets. Our implementation allows the quantum Fourier transform (QFT) and Shor's algorithm to be performed on a $D$-dimensional lattice in time logarithmic in the number of qubits for interactions with $\alpha \le D$. As a corollary, we show that power-law systems with $\alpha \le D$ are difficult to simulate classically even for short times, under a standard assumption that factoring is classically intractable. Complementarily, we develop a new technique to give a general lower bound, linear in the size of the system, on the time required to implement the QFT and the fanout gate in systems that are constrained by a linear light cone. This allows us to prove an asymptotically tighter lower bound for long-range systems than is possible with previously available techniques.Comment: 6 pages, 1 figur