20 pages, 9 figures, STACS'2003. v3: rewritten from scratch, new co-author, everything put into constant depth (including quantum Fourier transform). v4: polished a lot
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.
Recently, Bravyi, Gosset, and König (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 …
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 $1/r^α$ 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 t…
We first show how to construct an O(n)-depth O(n)-size quantum circuit for addition of two n-bit binary numbers with no ancillary qubits. The exact size is 7n-6, which is smaller than that of any other quantum circuit ever constructed for addition with no ancillary qubits. Using the circuit, we then propose a method for constructing an O(d(n))-depth O(n)-size quantum circuit for addition with O(n/d(n)) ancillary qubits for any d(n)=Ω(log n). If we are allowed to use unbounded fan-out gates with length O(n^c) for an arbitrary small positive constant c, we can modify the method and construct an …
Unconditional Quantum Advantage for Sampling with Shallow Circuits
Institute for Quantum Computing, University of Waterloo（滑铁卢大学量子计算研究所）；Department of Computer Science, Columbia University（哥伦比亚大学计算机科学系）；Perimeter Institute for Theoretical Physics（理论物理珀蒂默研究所）
We propose definitions of QAC^0, the quantum analog of the classical class AC^0 of constant-depth circuits with AND and OR gates of arbitrary fan-in, and QACC^0[q], where n-ary Mod-q gates are also allowed. We show that it is possible to make a `cat' state on n qubits in constant depth if and only if we can construct a parity or Mod-2 gate in constant depth; therefore, any circuit class that can fan out a qubit to n copies in constant depth also includes QACC^0[2]. In addition, we prove the somewhat surprising result that parity or fanout allows us to construct Mod-q gates in constant depth fo…
We show that the quantum parity gate on $n > 3$ qubits cannot be cleanly simulated by a quantum circuit with two layers of arbitrary C-SIGN gates of any arity and arbitrary 1-qubit unitary gates, regardless of the number of allowed ancilla qubits. This is the best known and first nontrivial separation between the parity gate and circuits of this form. The same bounds also apply to the quantum fanout gate. Our results are incomparable with those of Fang et al. [3], which apply to any constant depth but require a sublinear number of ancilla qubits on the simulating circuit.
Constant-Depth Quantum Imaginary Time Evolution Using Dynamic Fan-out Circuits
Jeppesen ForeFlight（杰普森福飞）；Department of Microtechnology and Nanoscience (MC2), Chalmers University of Technology（查尔姆斯理工大学微技术和纳米科学系（MC2））；Department of Chemistry and Chemical Engineering, Chalmers University of Technology（查尔姆斯理工大学化学与化学工程系）；Center for Advanced Systems Understanding, Helmholtz-Zentrum Dresden-Rossendorf（德累斯顿-罗斯多夫亥姆霍兹中心高级系统理解中心）；Center for Scalable Data Analytics and Artificial Intelligence Dresden/Leipzig, TU Dresden（德累斯顿/莱比锡可扩展数据分析与人工智能中心，德累斯顿工业大学）
We prove several new lower bounds for constant depth quantum circuits. The main result is that parity (and hence fanout) requires log depth circuits, when the circuits are composed of single qubit and arbitrary size Toffoli gates, and when they use only constantly many ancillæ. Under this constraint, this bound is close to optimal. In the case of a non-constant number $a$ of ancillae, we give a tradeoff between $a$ and the required depth, that results in a non-trivial lower bound for fanout when $a = n^{1-o(1)}$.
While quantum information processing by nuclear magnetic resonance (NMR) with small number of qubits is well established, implementation of lengthy computations have proved to be difficult due to decoherence/relaxation. In such circumstances, shallow circuits (circuits using parallel computation) may prove to be realistic. Parity and fanout gates are essential to create shallow circuits. In this article we implement inversion-on-equality gate, followed by parity gate and fanout gate in 3-qubit systems by NMR, using evolution under indirect exchange coupling Hamiltonian.
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 overlap. However, on many quantum hardware platforms, instructions on overlapping qubits can be executed simultaneously through __global interactions__. For example, while fan-out in traditional quantum circuits can only be implemented sequentially when viewed at the logical level, global interactions at the physical level allow fan-out to be achieved…
The Power of Shallow-depth Toffoli and Qudit Quantum Circuits
Sorbonne Université（索邦大学）；CNRS（法国国家科学研究中心）；School of Informatics, University of Edinburgh（爱丁堡大学信息学院）；International Iberian Nanotechnology Laboratory（伊比利亚纳米技术国际实验室）
We show that the depth of quantum circuits in the realistic architecture where a classical controller determines which local interactions to apply on the kD grid Z^k where k >= 2 is the same (up to a constant factor) as in the standard model where arbitrary interactions are allowed. This allows minimum-depth circuits (up to a constant factor) for the nearest-neighbor architecture to be obtained from minimum-depth circuits in the standard abstract model. Our work therefore justifies the standard assumption that interactions can be performed between arbitrary pairs of qubits. In particular, our …
Recently Bravyi, Gosset and König (Science 2018) proved an unconditional separation between the computational powers of small-depth quantum and classical circuits for a relation. In this paper we show a similar separation in the average-case setting that gives stronger evidence of the superiority of small-depth quantum computation: we construct a computational task that can be solved on all inputs by a quantum circuit of constant depth with bounded-fanin gates (a "shallow" quantum circuit) and show that any classical circuit with bounded-fanin gates solving this problem on a non-negligible fra…
Magneto-optical isolators and circulators have been widely used to safeguard quantum devices from reflections and noise in the readout stage. However, these devices have limited bandwidth, low tunability, are bulky, and suffer from high losses, making them incompatible with planar technologies such as circuit QED. To address these limitations, we propose a new approach to quantum non-reciprocity that utilizes the intrinsic nonlinearity of qubits and broken spatial symmetry. We show that a circuit containing Lorentz-type qubits can be transformed into Fano-type qubits with an asymmetric spectra…
The quantum query complexity of evaluating any read-once formula with n black-box input bits is Theta(sqrt(n)). However, the corresponding problem for read-many formulas (i.e., formulas in which the inputs have fanout) is not well understood. Although the optimal read-once formula evaluation algorithm can be applied to any formula, it can be suboptimal if the inputs have large fanout. We give an algorithm for evaluating any formula with n inputs, size S, and G gates using O(min{n, sqrt{S}, n^{1/2} G^{1/4}}) quantum queries. Furthermore, we show that this algorithm is optimal, since for any n,S…
We define a class of stochastic processes based on evolutions and measurements of quantum systems, and consider the complexity of predicting their long-term behavior. It is shown that a very general class of decision problems regarding these stochastic processes can be efficiently solved classically in the space-bounded case. The following corollaries are implied by our main result: (1) Any space O(s) uniform family of quantum circuits acting on s qubits and consisting of unitary gates and measurement gates defined in a typical way by matrices of algebraic numbers can be simulated by an unboun…
We prove that one-way quantum computations have the same computational power as quantum circuits with unbounded fan-out. It demonstrates that the one-way model is not only one of the most promising models of physical realisation, but also a very powerful model of quantum computation. It confirms and completes previous results which have pointed out, for some specific problems, a depth separation between the one-way model and the quantum circuit model. Since one-way model has the same computational power as unbounded quantum fan-out circuits, the quantum Fourier transform can be approximated in…
Distinguishing logarithmic depth quantum circuits on mixed states is shown to be complete for QIP, the class of problems having quantum interactive proof systems. Circuits in this model can represent arbitrary quantum processes, and thus this result has implications for the verification of implementations of quantum algorithms. The distinguishability problem is also complete for QIP on constant depth circuits containing the unbounded fan-out gate. These results are shown by reducing a QIP-complete problem to a logarithmic depth version of itself using a parallelization technique.
We prove that constant-depth quantum circuits are more powerful than their classical counterparts. To this end we introduce a non-oracular version of the Bernstein-Vazirani problem which we call the 2D Hidden Linear Function problem. An instance of the problem is specified by a quadratic form q that maps n-bit strings to integers modulo four. The goal is to identify a linear boolean function which describes the action of q on a certain subset of n-bit strings. We prove that any classical probabilistic circuit composed of bounded fan-in gates that solves the 2D Hidden Linear Function problem wi…
Worst-case depth hierarchy for shallow quantum circuits
Hon Hai (Foxconn) Quantum Computing Research Center（鸿海（富士康）量子计算研究中心）；University of Oxford（牛津大学）；The University of Hong Kong（香港大学）；University of Technology Sydney（悉尼科技大学）
We propose definitions of $\QAC^0$, the quantum analog of the classical class $\AC^0$ of constant-depth circuits with AND and OR gates of arbitrary fan-in, and $\QACC[q]$, the analog of the class $\ACC[q]$ where $\Mod_q$ gates are also allowed. We prove that parity or fanout allows us to construct quantum $\MOD_q$ gates in constant depth for any $q$, so $\QACC[2] = \QACC$. More generally, we show that for any $q,p > 1$, $\MOD_q$ is equivalent to $\MOD_p$ (up to constant depth). This implies that $\QAC^0$ with unbounded fanout gates, denoted $\QACwf^0$, is the same as $\QACC[q]$ and $\QACC$ for…
Recent work by Bravyi et al. constructs a relation problem that a noisy constant-depth quantum circuit (QNC$^0$) can solve with near certainty (probability $1 - o(1)$), but that any bounded fan-in constant-depth classical circuit (NC$^0$) fails with some constant probability. We show that this robustness to noise can be achieved in the other low-depth quantum/classical circuit separations in this area. In particular, we show a general strategy for adding noise tolerance to the interactive protocols of Grier and Schaeffer. As a consequence, we obtain an unconditional separation between noisy QN…
Let $η_0$ be the supremum of those $η$ for which every poly-size quantum circuit can be simulated by another poly-size quantum circuit with gates of fan-in $\leq 2$ that tolerates random noise independently occurring on all wires at the constant rate $η$. Recent fundamental results showing the principal fact $η_0>0$ give estimates like $η_0\geq 10^{-6}-10^{-4}$, whereas the only upper bound known before is $η_0\leq 0.74$. In this note we improve the latter bound to $η_0\leq 1/2$, under the assumption $QP\not\subseteq QNC^1$. More generally, we show that if the decoherence rate $η$ is greater t…
In breakthrough work, Bravyi, Gosset, and König (BGK) [Science, 2018] unconditionally proved that constant depth quantum circuits are more powerful than their classical counterparts. Their result is equivalent to saying that a particular family of constant depth quantum circuits takes classical circuits at least $Ω(\log n)$ depth to "simulate", in a certain sense. In our paper, we formalise their sense of simulation, which we call "possibilistic simulation" or "p-simulation", and construct explicit classical circuits that can p-simulate any depth-$d$ quantum circuit with Clifford and $t$ $T$-g…
Low depth measurement-based quantum computation with qudits ($d$-level systems) is investigated and a precise relationship between this powerful model and qudit quantum circuits is derived in terms of computational depth and size complexity. To facilitate this investigation a qudit `unbounded fan-out' circuit model, in which a qudit may be quantum-copied into an arbitrary number of ancillas in a single time-step, is introduced and shown to be capable of implementing interesting $n$-qudit unitaries in constant depth. A procedure for reducing the quantum computational depth in the measurement-ba…
This paper shows that, in measurement-based quantum computation, it is possible to write any quantum oracle implementing a classical function in constant depth. The result is shown through the equivalence between MBQC and the circuit model where arbitrary rotations along $Z$ axis and unbounded fan-outs are elementary operations. A corollary of this result is that disjunction can be implemented exactly in constant-depth, answering an open question of Høyer and Špalek.
Measurement-driven quantum advantages in shallow circuits
Dahlem Center for Complex Quantum Systems, Freie Universität Berlin（柏林自由大学达勒姆复杂量子系统中心）；Helmholtz-Zentrum Berlin für Materialien und Energie（柏林亥姆霍兹材料能源中心）
Through the two specific problems, the 2D hidden linear function problem and the 1D magic square problem, Bravyi et al. have recently shown that there exists a separation between $\mathbf{QNC^0}$ and $\mathbf{NC^0}$, where $\mathbf{QNC^0}$ and $\mathbf{NC^0}$ are the classes of polynomial-size and constant-depth quantum and classical circuits with bounded fan-in gates, respectively. In this paper, we present another problem with the same property, the magic pentagram problem based on the magic pentagram game, which is a nonlocal game. In other words, we show that the problem can be solved with…
We present two results on the subject of quantum contextuality and cohomology, and non-locality and quantum advantage with shallow circuits. Abramsky et al. showed that a range of examples of quantum contextuality is detected by a cohomological invariant based on Čech cohomology. However, the approach does not give a complete cohomological characterisation of contextuality. A different cohomological approach to contextuality was introduced by Okay et al. Their approach exploits the algebraic structure of the Pauli operators and their qudit generalisations known as Weyl operators. We give an ab…
The quantum Fourier transform (QFT) is the principal algorithmic tool underlying most efficient quantum algorithms. We present a generic framework for the construction of efficient quantum circuits for the QFT by ``quantizing'' the separation of variables technique that has been so successful in the study of classical Fourier transform computations. Specifically, this framework applies the existence of computable Bratteli diagrams, adapted factorizations, and Gel'fand-Tsetlin bases to offer efficient quantum circuits for the QFT over a wide variety a finite Abelian and non-Abelian groups, incl…
An unconditional distribution learning advantage with shallow quantum circuits
Technische Universität Berlin（柏林工业大学）；Freie Universität Berlin（柏林自由大学）；Fraunhofer Heinrich Hertz Institute（弗劳恩霍夫海因里希·赫兹研究所）；Helmholtz-Zentrum Berlin für Materialien und Energie（亥姆霍兹柏林材料与能源研究中心）
We provide exact results for the dynamics of local-operator entanglement in quantum circuits with two-dimensional wires featuring ultralocal solitons, i.e. single-site operators which, up to a phase, are simply shifted by the time evolution. We classify all circuits allowing for ultralocal solitons and show that only dual-unitary circuits can feature moving ultralocal solitons. Then, we rigorously prove that if a circuit has an ultralocal soliton moving to the left (right), the entanglement of local operators initially supported on even (odd) sites saturates to a constant value and its dynamic…
We contribute a 2D nearest-neighbor quantum architecture for Shor's algorithm to factor an $n$-bit number in $O(\log^2(n))$ depth. Our implementation uses parallel phase estimation, constant-depth fanout and teleportation, and constant-depth carry-save modular addition. We derive upper bounds on the circuit resources of our architecture under a new 2D nearest-neighbor model which allows a classical controller and parallel, communicating modules. We also contribute a novel constant-depth circuit for unbounded quantum unfanout in our new model. Finally, we provide a comparison to all previous ne…
Explicit channels with unbounded gains in classical communication using entangled inputs
National Taiwan University（国立台湾大学）；National Center for Theoretical Sciences, Taiwan（台湾理论科学中心）；Hon Hai (Foxconn) Quantum Computing Center, Taiwan（台湾鸿海（富士康）量子计算中心）；Concordia University（康考迪亚大学）；Syracuse University（雪城大学）
