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arXiv:1412.1349 · Physical Review Letters

Universal Superreplication of Unitary Gates

Giulio Chiribella, Yuxiang Yang, C. Huang

Published Mar 25, 2015• 26 citations
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Abstract

Quantum states obey an asymptotic no-cloning theorem, stating that no deterministic machine can reliably replicate generic sequences of identically prepared pure states. In stark contrast, we show that generic sequences of unitary gates can be replicated deterministically at nearly quadratic rates, with an error vanishing on most inputs except for an exponentially small fraction. The result is not in contradiction with the no-cloning theorem, since the impossibility of deterministically transforming pure states into unitary gates prevents the application of the gate replication protocol to states. In addition to gate replication, we show that N parallel uses of a completely unknown unitary gate can be compressed into a single gate acting on O(log_{2}N) qubits, leading to an exponential reduction of the amount of quantum communication needed to implement the gate remotely.

DOI
10.1103/physrevlett.114.120504

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Quantum states obey an asymptotic no-cloning theorem, stating that no deterministic machine can reliably replicate generic sequences of identically prepared pure states. In stark contrast, we show that generic sequences of unitary gates can be replicated deterministically at nearly quadratic rates, with an error vanishing on most inputs except for an exponentially small fraction. The result is not in contradiction with the no-cloning theorem, since the impossibility of deterministically transforming pure states into unitary gates prevents the application of the gate replication protocol to states. In addition to gate replication, we show that N parallel uses of a completely unknown unitary gate can be compressed into a single gate acting on O(log_{2}N) qubits, leading to an exponential reduction of the amount of quantum communication needed to implement the gate remotely.
Although the no-cloning theorem forbids perfect replication of quantum information, it is sometimes possible to produce large numbers of replicas with vanishingly small error. This phenomenon, known as quantum superreplication, can occur for both quantum states and quantum gates. The aim of this paper is to review the central features of quantum superreplication and provide a unified view of existing results. The paper also includes new results. In particular, we show that when quantum superreplication can be achieved, it can be achieved through estimation up to an error of size O(M/N 2), where N and M are the number of input and output copies, respectively. Quantum strategies still offer an advantage for superreplication in that they allow for exponentially faster reduction of the error. Using the relation with estimation, we provide i) an alternative proof of the optimality of Heisenberg scaling in quantum metrology, ii) a strategy for estimating arbitrary unitary gates with a mean square error scaling as log N/N 2, and iii) a protocol that generates O(N 2) nearly perfect copies of a generic pure state U |0〉 while using the corresponding gate U only N times. Finally, we point out that superreplication can be achieved using interactions among k systems, provided that k is large compared to M 2/N 2.
We show that one can deterministically generate, out of N copies of an unknown unitary operation, up to N^{2} almost perfect copies. The result holds for all operations generated by a Hamiltonian with an unknown interaction strength. This generalizes a similar result in the context of phase-covariant cloning where, however, superreplication comes at the price of an exponentially reduced probability of success. We also show that multiple copies of unitary operations can be emulated by operations acting on a much smaller space, e.g., a magnetic field acting on a single n-level system allows one to emulate the action of the field on n^{2} qubits.
It is known that the classical information like strings of bits can be copied. In 1982, Wootters and Zurek proposed the quantum no-cloning principle. No-cloning principle says that it is impossible to make an identical copy of an arbitrary unknown quantum pure state by using unitary evolution. In this paper, we call U ∣ψ>|ψ>∣ψ>, where U is any unitary operator, a U-copy of the state ∣ψ>|ψ>∣ψ> and show it is impossible to make a U-copy of an arbitrary unknown quantum pure state by using unitary evolution.
We consider the problem of deterministically cloning quantum channels with respect to the best attainable rate and the highest quality, so-called optimal cloning. We demonstrate that cloning quantum states is, in-fact, equivalent to cloning the trash-and-replace channel and therefore the former is a special case of the more general problem. By appealing to higher-order quantum operations (quantum processes) we construct a unified framework to deal with the most general cloning tasks and establish necessary conditions for a family of channels to exhibit super-replication -- a quadratic cloning rate with vanishing error. We find that noisy phase-gate channels satisfy these conditions, and we construct the explicit super-replicating process for the task. Conversely, we find that the criteria are not met by the full set of noisy unitary gates; classical noise channels; or amplitude damping channels, whose respective cloning rates are at most linear. In this paradigm, we not only derive new results, but also refigure known ones. We derive a strong converse for state cloning, and for unitary channels we construct an alternative super-replication process to that of Dür et al. [PRL 114, 120503 (2015)] and Chiribella et al. [PRL 114, 120504 (2015)] based on a measure-and-prepare process, which allows us to establish a direct connection between optimal channel cloning and Bayesian channel estimation. Finally we give an SDP algorithm to search for optimal cloning processes and study the advantage of coherent vs measure-and-prepare protocols on concrete examples.
We experimentally demonstrate the underlying physical mechanism of the recently proposed protocol for superreplication of quantum phase gates [W. D\"ur, P. Sekatski, and M. Skotiniotis, Phys. Rev. Lett. 114, 120503 (2015)], which allows producing up to N2{N}^{2}N2 high-fidelity replicas from NNN input copies in the limit of large NNN. Our implementation of $1\ensuremath{\rightarrow}2$ replication of the single-qubit phase gates is based on linear optics and qubits encoded into states of single photons. We employ the quantum Toffoli gate to imprint information about the structure of an input two-qubit state onto an auxiliary qubit, apply the replicated operation to the auxiliary qubit, and then disentangle the auxiliary qubit from the other qubits by a suitable quantum measurement. We characterize the replication protocol by full quantum process tomography and observe good agreement of the experimental results with theory.
The no-cloning theorem is one of the fundamental concepts of quantum information theory. It tells us that no general quantum state can be perfectly replicated. In this thesis we introduce the notion of imperfect cloning, and define the properties of the universal cloning machine. Furthermore, we construct an ansatz of how our universal cloning machine should perform as to produce two imperfect clones from one input qubit. We find an optimal fidelity of 5/6 for our universal cloning machine. We then reevaluate our ansatz and construct a class of unitary transformations such that an optimal fidelity is always achieved. Lastly, we present an overview of some applications of imperfect quantum cloning in the field of quantum information.