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Quantum computation offers the potential to solve fundamental yet otherwise intractable problems across a range of active fields of research. Recently, universal quantum-logic gate sets - the building blocks for a quantum computer - have been demonstrated in several physical architectures. A serious obstacle to a full-scale implementation is the sheer number of these gates required to implement even small quantum algorithms. Here we present and demonstrate a general technique that harnesses higher dimensions of quantum systems to significantly reduce this number, allowing the construction of key quantum circuits with existing technology. We are thereby able to present the first implementation of two key quantum circuits: the three-qubit Toffoli and the two-qubit controlled-unitary. The gates are realised in a linear optical architecture, which would otherwise be absolutely infeasible with current technology.
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This paper extends the quantum search class of algorithms to the multiple solution case. It is shown that, like the basic search algorithm, these too can be represented as a rotation in an appropriately defined two dimensional vector space. This yields new applications - an algorithm is presented that can create an arbitrarily specified quantum superposition on a space of size N in O(sqrt(N)) steps. By making a measurement on this superposition, it is possible to obtain a sample according to an arbitrarily specified classical probability distribution in O(sqrt(N)) steps. A classical algorithm would need O(N) steps.
We construct a three-dimensional quantum cellular automaton (QCA), an automorphism of the local operator algebra on a lattice of qubits, which disentangles the ground state of the Walker-Wang three fermion model. We show that if this QCA can be realized by a quantum circuit of constant depth, then there exists a two-dimensional commuting projector Hamiltonian which realizes the three fermion topological order which is widely believed not to be possible. We conjecture in accordance with this belief that this QCA is not a quantum circuit of constant depth, and we provide two further pieces of evidence to support the conjecture. We show that this QCA maps every local Pauli operator to a local Pauli operator, but is not a Clifford circuit of constant depth. Further, we show that if the three-dimensional QCA can be realized by a quantum circuit of constant depth, then there exists a two-dimensional QCA acting on fermionic degrees of freedom which cannot be realized by a quantum circuit of constant depth; i.e., we prove the existence of a nontrivial QCA in either three or two dimensions. The square of our three-dimensional QCA can be realized by a quantum circuit of constant depth, and this suggests the existence of a $mathbb{Z}_2$ invariant of a QCA in higher dimensions, totally distinct from the classification by positive rationals (i.e., by one integer index for each prime) in one dimension. In an appendix, unrelated to the main body of this paper, we give a fermionic generalization of a result of Bravyi and Vyalyi on ground states of 2-local commuting Hamiltonians.
The Clifford hierarchy is a nested sequence of sets of quantum gates critical to achieving fault-tolerant quantum computation. Diagonal gates of the Clifford hierarchy and nearly diagonal semi-Clifford gates are particularly important: they admit efficient gate teleportation protocols that implement these gates with fewer ancillary quantum resources such as magic states. Despite the practical importance of these sets of gates, many questions about their structure remain open; this is especially true in the higher-dimensional qudit setting. Our contribution is to leverage the discrete Stone-von Neumann theorem and the symplectic formalism of qudit stabiliser mechanics towards extending results of Zeng-Cheng-Chuang (2008) and Beigi-Shor (2010) to higher dimensions in a uniform manner. We further give a simple algorithm for recursively enumerating all gates of the Clifford hierarchy, a simple algorithm for recognising and diagonalising semi-Clifford gates, and a concise proof of the classification of the diagonal Clifford hierarchy gates due to Cui-Gottesman-Krishna (2016) for the single-qudit case. We generalise the efficient gate teleportation protocols of semi-Clifford gates to the qudit setting and prove that every third level gate of one qudit (of any prime dimension) and of two qutrits can be implemented efficiently. Numerical evidence gathered via the aforementioned algorithms support the conjecture that higher-level gates can be implemented efficiently.
Thinning antenna arrays through quantum Fourier transform (QFT) is proposed. Given the lattice of the candidate locations for the array elements, the problem of selecting which antenna location has to be either occupied or not by an array element is formulated in the quantum computing (QC) framework and then addressed with an ad-hoc design method based on a suitable implementation of the QFT algorithm. Representative numerical results are presented and discussed to point out the features and the advantages of the proposed QC-based thinning technique.
In search of a quantum key distribution scheme that could stand up for more drastic eavesdropping attack, I discover a prepare-and-measure scheme using $N$-dimensional quantum particles as information carriers where $N$ is a prime power. Using the Shor-Preskill-type argument, I prove that this scheme is unconditional secure against all attacks allowed by the laws of quantum physics. Incidentally, for $N = 2^n > 2$, each information carrier can be replaced by $n$ entangled qubits. And in this case, I discover an eavesdropping attack on which no unentangled-qubit-based prepare-and-measure quantum key distribution scheme known to date can generate a provably secure key. In contrast, this entangled-qubit-based scheme produces a provably secure key under the same eavesdropping attack whenever $N geq 16$. This demonstrates the advantage of using entangled particles as information carriers to combat certain eavesdropping strategies.
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