Universal Superreplication of Unitary Gates

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 N ) qubits, leading to an exponential reduction of the amount of quantum communication needed to implement the gate remotely.A striking feature of quantum theory is the impossibility of constructing a universal copy machine, which takes as input a quantum system in an arbitrary pure state and produces as output a number of exact replicas [1,2]. Such an impossibility has major implications for quantum error correction [3,4] and cryptographic protocols such as key distribution [5,6], quantum secret sharing [7,8], and quantum money [9][10][11][12].The impossibility of universal copy machines is a hard fact: it equally affects deterministic [13][14][15][16] and probabilistic machines [17][18][19], whose performances coincide with those of deterministic machines when it comes to copying completely unknown pure states [18,19]. A similar no-go result holds when the universal machine is presented a large number N of identical copies and is required to produce a larger number M > N of approximate replicas: if the replicas have non-vanishing overlap with the desired M -copy state, then the number of extra copies must be negligible compared to N [15]. We refer to this fact as the asymptotic no-cloning theorem, expressing the fact that independent and identically distributed (i.i.d.) sequences of pure states cannot be stretched by any significant amount. The asymptotic no-cloning theorem holds also for non-universal machines designed to copy continuous sets of states, provided that such machines work deterministically [19].The impossibility of universal state cloning suggests similar results for quantum gates. Along this line, a nogo theorem for universal gate cloning was proven in Ref. [20], showing that no quantum network can perfectly simulate two uses of an unknown unitary gate by querying it only once. Optimal networks that approximate universal gate cloning were studied in Refs. [20,21]. Very recently, Dür and coauthors [22] considered a non-universal setup designed to clone phase gates, i. e. gates generated by time evolution with a known Hamiltonian. In this scenario, they devised a quantum network that approximately simulates up to N 2 uses of an unknown phase gate while using it only N times, with vanishing error in the large N limit. Such a result establishes the possibi...

Experimental replication of single-qubit quantum phase gates

“…4 for four values of the phase shift φ: 0, π 2 , π, and 3π 2 . Additional results are provided in the Appendix which for comparison also contains plots of the corresponding theoretical matrices χ th of the ideal two-qubit unitary operations (8). We quantify the performance of our experimental scheme by the quantum process fidelity F CU of each implemented operation with the corresponding unitary operation CU (φ) that would be implemented by our setup under ideal experimental conditions.…”

“…Recently, the concept of quantum cloning was extended from states to quantum operations [5,6], which brings in new interesting aspects and features. In particular, it was shown very recently that starting from N copies of a single-qubit unitary operation U one can deterministically generate up to N 2 copies of this operation with an exponentially small error [7,8]. As pointed out in Ref.…”

“…(8) shows that the quantum circuit depicted in Fig. 1(b) can be also interpreted as a scheme for deterministic conversion of an arbitrary single-qubit unitary phase gate U (φ) into a two-qubit controlled unitary gate CU (φ) [10][11][12].…”

Quantum-enhanced learning of rotations about an unknown direction

“…Its overall structure is summarised in figure 2. The temporal separation between the training phase and the execution phase makes the U x -to-V x learning problem distinct from the problem of simulating the gate V x using the gate U x as a resource [30][31][32][33][34][35]. In that problem, the gate V x is simulated by an arbitrary circuit using the gate U x , not necessarily a circuit of the form depicted in figure 2.…”

Quantum compression of tensor network states

“…Universal compression protocols for the scenario where the average state of each system is unknown, except for an upper bound on its von Neumann entropy, were provided in [8]. In recent years, there has been an interest in developing compression protocols for identically prepared systems [9][10][11][12][13]. Such systems occur in a wide range of tasks, including quantum tomography [14,15], quantum cloning [16,17], estimation [18,19], and quantum machine learning [20].…”