With a combination of the quantum repeater and the cluster state approaches, we show that efficient quantum computation can be constructed even if all the entangling quantum gates only succeed with an arbitrarily small probability $p$. The required computational overhead scales efficiently both with $1/p$ and $n$, where $n$ is the number of qubits in the computation. This approach provides an efficient way to combat noise in a class of quantum computation implementation schemes, where the dominant noise leads to probabilistic signaled errors with an error probability $1-p$ far beyond any threshold requirement.
Nonlocal gate operation is based on sharing an ancillary pair of qubits in perfect entanglement. When the ancillary pair are partially entangled, the efficiency of the gate operation drops. Using general transformations, we devise probabilistic nonlocal gates, which perform the nonlocal operation conclusively when the ancillary pair are only partially entangled. We show that a controlled purification protocol can be implemented by the probabilistic nonlocal operation.
We propose a scheme to perform probabilistic quantum gates on remote trapped atom qubits through interference of optical frequency qubits. The method does not require localization of the atoms to the Lamb-Dicke limit, and is not sensitive to interferometer phase instabilities. Such probabilistic gates can be used for scalable quantum computation.
We propose a scalable way to construct a 3D cluster state for fault-tolerant topological one-way computation (TOWC) even if the entangling two-qubit gates succeed with a small probability. It is shown that fault-tolerant TOWC can be performed with the success probability of the two-qubit gate such as 0.5 (0.1) provided that the conditional error probability of the two-qubit gate is less than 0.040% (0.016%). Furthermore, the resource usage is considerably suppressed compared to the conventional fault-tolerant schemes with probabilistic two-qubit gates.
We propose an efficient scheme for constructing arbitrary 2-D cluster states using probabilistic entangling quantum gates.In our scheme, the 2-D cluster state is constructed with star-like basic units generated from 1-D cluster chains.By applying parallel operations, the process of generating 2-D (or higher dimensional) cluster states is significantly accelerated, which provides an efficient way to implement realistic one way quantum computers.
We show that the Hadamard and Unitary gates could be implemented by a unitary evolution together with a measurement for any unknown state chosen from a set $A = {{| {Ψ_i} > ,| {\bar Ψ_i} >}}({i = 1,2})$ if and only if $| {Ψ_1} > ,| {Ψ_2} > ,| {\bar Ψ_1} > ,| {\bar Ψ_2} >$ are linearly independent. We also derive the best transformation efficiencies.
A new approach to efficient quantum computation with probabilistic gates is proposed and analyzed in both a local and non-local setting. It combines heralded gates previously studied for atom or atom-like qubits with logical encoding from linear optical quantum computation in order to perform high fidelity quantum gates across a quantum network. The error-detecting properties of the heralded operations ensure high fidelity while the encoding makes it possible to correct for failed attempts such that deterministic and high-quality gates can be achieved. Importantly, this is robust to photon los…
Probabilistic storage and retrieval (PSR) of unitary quantum dynamics is possible with exponentially small failure probability with respect to the number of systems used as a quantum memory [PRL 122, 170502 (2019)]. Here we study improvements due to a priori knowledge about the unitary transformation to be stored. In particular, we study $N \rightarrow 1$ PSR of qubit phase gates, i.e. qubit rotations a round $Z$ axis with an unknown angle, and show that if we access the gate only $N$-times, the optimal probability of perfect retrieving of its single use is $N/(N+1)$. We propose a quantum circ…
Simulation of quantum matters is a significant application of quantum computers. In contrast to the unitary operation which can be realized naturally on a quantum computer, the implementation of nonunitary operation, widely used in classical approaches, needs special designing. Here, by application of Grover's algorithm, we extend the probabilistic method of implementing nonunitary operation and show that it can promote success probability without fidelity decreasing. This method can be applied to problems of imaginary time evolution and contraction of tensor networks on a quantum computer.
We establish bounds to the necessary resource consumption when building up cluster states for one-way computing using probabilistic gates. Emphasis is put on state preparation with linear optical gates, as the probabilistic character is unavoidable here. We identify rigorous general bounds to the necessary consumption of initially available maximally entangled pairs when building up one-dimensional cluster states with individually acting linear optical quantum gates, entangled pairs and vacuum modes. As the known linear optics gates have a limited maximum success probability, as we show, this …
We propose probabilistic controlled-NOT and controlled-phase gates for qubits stored in the polarization of photons. The gates are composed of linear optics and photon detectors, and consume polarization entangled photon pairs. The fraction of the successful operation is only limited by the efficiency of the Bell-state measurement. The gates work correctly under the use of imperfect detectors and lossy transmission of photons. Combined with single-qubit gates, they can be used for producing arbitrary polarization states and for designing various quantum measurements.
Probabilistic quantum cloning and identifying machines can be constructed via unitary-reduction processes [Duan and Guo, Phys. Rev. Lett. 80, 4999 (1998)]. Given the cloning (identifying) probabilities, we derive an explicit representation of the unitary evolution and corresponding Hamiltonian to realize probabilistic cloning (identification). The logic networks are obtained by decomposing the unitary representation into universal quantum logic operations. The robustness of the networks is also discussed. Our method is suitable for a $k$-partite system, such as quantum computer, and may be gen…
A simplification scheme of probabilistic teleportation of two-particle state of general form is given. By means of the primitive operations consisting of single-qubit gates, two-qubit controlled-not gates, Von Neumann measurement and classically controlled operations, we construct an efficient quantum logical network for implementing the new scheme of probabilistic teleportation of a two-particle state of general form.
The scheme for probabilistic teleportation of an N-particle state of general form is proposed. As the special cases we construct efficient quantum logic networks for implementing probabilistic teleportation of a two-particle state, a three-particle state and a four-particle state of general form, built from single qubit gates, two-qubit controlled-not gates, Von Neumann measurement and classically controlled operations.
We propose a method for scaling trapped ions for large-scale quantum computation and communication based on a probabilistic ion-photon mapping. Deterministic quantum gates between remotely located trapped ions can be achieved through detection of spontaneously-emitted photons, accompanied by the local Coulomb interaction between neighboring ions. We discuss gate speeds and tolerance to experimental noise for different probabilistic entanglement schemes.
We propose a new measure of non-classicality of quantum gates which is particularly suitable for probabilistic devices. This measure enables to compare, e.g., deterministic devices which prepare entangled states with low amount of entanglement with probabilistic devices which generate highly entangled states but which fail sometimes. We provide examples demonstrating advantages of this new measure over the so far employed entangling power.
We present simplification schemes for probabilistic and controlled teleportation of the unknown quantum states of both one-particle and two-particle and construct efficient quantum logic networks for implementing the new schemes by means of the primitive operations consisting of single-qubit gates, two-qubit controlled-not gates, Von Neumann measurement and classically controlled operations. In these schemes the teleportation are not always successful but with certain probability.
A probabilistic imaginary-time evolution quantum algorithm for advection-diffusion equation: Explicit gate-level implementation and comparisons to quantum linear system algorithms
It is shown that the fault testing for quantum circuits does not follow conventional classical techniques. If probabilistic gate like Hadamard gate is included in a circuit then the classical notion of test vector is shown to fail. We have reported several new and distinguishing features of quantum fault and also presented a general methodology for detection of functional faults in a quantum circuit. The technique can generate test vectors for detection of different kinds of fault. Specific examples are given and time complexity of the proposed quantum fault detection algorithm is reported.
We clarify the connections between the erasure scheme of probabilistic CNOT gate implementation recently proposed by Pittman, Jacobs and Franson [Phys. Rev. A 64, 062311 (2001)] and quantum teleportation.
We assess the resources needed to identify a reversible quantum gate among a finite set of alternatives, including in our analysis both deterministic and probabilistic strategies. Among the probabilistic strategies we consider unambiguous gate discrimination, where errors are not tolerated but inconclusive outcomes are allowed, and we prove that parallel strategies are sufficient to unambiguously identify the unknown gate with minimum number of queries. This result is used to provide upper and lower bounds on the query complexity and on the minimum ancilla dimension. In addition, we introduce …
It has previously been shown that probabilistic quantum logic operations can be performed using linear optical elements, additional photons (ancilla), and post-selection based on the output of single-photon detectors. Here we describe the operation of several quantum logic operations of an elementary nature, including a quantum parity check and a quantum encoder, and we show how they can be combined to implement a controlled-NOT (CNOT) gate. All of these gates can be constructed using polarizing beam splitters that completely transmit one state of polarization and totally reflect the orthogona…
Recently it has been shown that Repeat-Until-Success (RUS) circuits can approximate a given single-qubit unitary with an expected number of $T$ gates of about $1/3$ of what is required by optimal, deterministic, ancilla-free decompositions over the Clifford+$T$ gate set. In this work, we introduce a more general and conceptually simpler circuit decomposition method that allows for synthesis into protocols that probabilistically implement quantum circuits over several universal gate sets including, but not restricted to, the Clifford+$T$ gate set. The protocol, which we call Probabilistic Quant…
We show how to construct quantum gate arrays that can be programmed to perform different unitary operations on a data register, depending on the input to some program register. It is shown that a universal quantum gate array - a gate array which can be programmed to perform any unitary operation - exists only if one allows the gate array to operate in a probabilistic fashion. The universal quantum gate array we construct requires an exponentially smaller number of gates than a classical universal gate array.
We demonstrate a probabilistic entangling quantum gate between two distant trapped ytterbium ions. The gate is implemented between the hyperfine "clock" state atomic qubits and mediated by the interference of two emitted photons carrying frequency encoded qubits. Heralded by the coincidence detection of these two photons, the gate has an average fidelity of 90+-2%. This entangling gate together with single qubit operations is sufficient to generate large entangled cluster states for scalable quantum computing.
Pure quantum states are often approximately encoded as classical bit strings such as those representing probability amplitudes and those describing circuits that generate the quantum states. The crucial quantity is the minimum length of classical bit strings from which the original pure states are approximately reconstructible. We derive asymptotically tight bounds on the minimum bit length required for probabilistic encodings with which one can approximately reconstruct the original pure state as an ensemble of the quantum states encoded in classical strings. We also show that such a probabil…
We realize the probabilistic cloning and identifying linear independent quantum states of multi-particles system, given prior probability, with universal quantum logic gates using the method of unitary representation. Our result is universal for separate state and entanglement. We also provide the realization in the condition given $M$ initial copies for each state.
We propose a new way of implementing several elementary quantum gates for qubits in the coherent state basis. The operations are probabilistic and employ single photon subtractions as the driving force. Our schemes for single-qubit phase gate and two-qubit controlled phase gate are capable of achieving arbitrarily large phase shifts with currently available resources, which makes them suitable for the near-future tests of quantum information processing with superposed coherent states.
We present a computational circuit which realizes contextually the Tele-UNOT gate and the universal optimal quantum cloning machine (UOQCM). We report the experimental realization of the probabilistic UOQCM with polarization encoded qubits. This is achieved by combining on a symmetric beam-splitter the input qubit with an ancilla in a fully mixed state.
Quantum controlled-logic gates, including controlled NOT gate and Toffoli gate, play critical roles in lots of quantum information processing schemes. We design and experimentally demonstrate deterministic Toffoli gate by utilizing orbital-angular-momentum and polarization degrees of freedom of a single photon. In addition, we generate Bell states by using the controlled NOT gate. The effective conversion rate of the Toffoli gate in our experiment is $(95.1\pm3.2)\%$. Furthermore, our experimental setup does not require any auxiliary photons and probabilistic post selections.
We propose a general framework for quantum error mitigation that combines and generalizes two techniques: probabilistic error cancellation (PEC) and zero-noise extrapolation (ZNE). Similarly to PEC, the proposed method represents ideal operations as linear combinations of noisy operations that are implementable on hardware. However, instead of assuming a fixed level of hardware noise, we extend the set of implementable operations by noise scaling. By construction, this method encompasses both PEC and ZNE as particular cases and allows us to investigate a larger set of hybrid techniques. For ex…
In a large-scale quantum computer, the cost of communications will dominate the performance and resource requirements, place many severe demands on the technology, and constrain the architecture. Unfortunately, fault-tolerant computers based entirely on photons with probabilistic gates, though equipped with "built-in" communication, have very large resource overheads; likewise, computers with reliable probabilistic gates between photons or quantum memories may lack sufficient communication resources in the presence of realistic optical losses. Here, we consider a compromise architecture, in wh…
Quantum teleportation is a powerful protocol with applications in several schemes of quantum communication, quantum cryptography and quantum computing. The present work shows the required conditions for a two-qubit quantum gate to be deterministically and probabilistically teleported by a quantum gate teleportation scheme using different basis of measurement. Additionally, we present examples of teleportation of two-qubit gates that do not belong to Clifford group as well the limitations of the quantum gate teleportation scheme employing a four-qubit state with genuine four-way entanglement.
We investigate robustness of probabilistic storage and retrieval device optimized for phase gates to noise. We use noisy input composed of convex combination of unitary channel with either depolarizing or dephasing channel. We find out that the resistance to dephasing noise is higher than to depolarization. Interestingly, for the depolarisation the retrieval reduces the degree of noise. We also examine the possible realizations showing that their performance is different when the noise is present.
The linear optics approach to quantum computing has several potential advantages but the logic operations are probabilistic. Here we review the use of the quantum Zeno effect to suppress the intrinsic failure events in these kinds of devices, which would produce deterministic logic operations without the need for ancilla photons or high-efficiency detectors. The potential advantages of implementing Zeno gates using micro-cavities and electromagnetically-induced transparency are discussed.
We discuss and experimentally demonstrate a probabilistic Hadamard gate for coherent state qubits. The scheme is based on linear optical components, non-classical resources and the joint projective action of a photon counter and a homodyne detector. We experimentally characterize the gate for the coherent states of the computational basis by full tomographic reconstruction of the transformed output states. Based on the parameters of the experiment we simulate the fidelity for all coherent state qubits on the Bloch sphere.
One of the main advantages of an optical approach to quantum computing is the fact that optical fibers can be used to connect the logic and memory devices to form useful circuits, in analogy with the wires of a conventional computer. Here we describe an experimental demonstration of a simple quantum circuit of that kind in which two probabilistic exclusive-OR (XOR) logic gates were combined to calculate the parity of three input qubits.
We address the question of a quantum memory storage of quantum dynamics. In particular, we design an optimal protocol for $N\to 1$ probabilistic storage-and-retrieval of unitary channels on $d$-dimensional quantum systems. If we may access the unknown unitary gate only $N$-times, the optimal success probability of perfect retrieval of its single use is $N/(N-1+d^2)$. The derived size of the memory system exponentially improves the known upper bound on the size of the program register needed for probabilistic programmable quantum processors. Our results are closely related to probabilistic perf…
An ideal controlled-NOT gate followed by projective measurements can be used to identify specific Bell states of its two input qubits. When the input qubits are each members of independent Bell states, these projective measurements can be used to swap the post-selected entanglement onto the remaining two qubits. Here we apply this strategy to produce heralded two-photon polarization entanglement using Bell states that originate from independent parametric down-conversion sources, and a particular probabilistic controlled-NOT gate that is constructed from linear optical elements. The resulting …
We design linear optics multiqubit quantum logic gates. We assume the traditional encoding of a qubit onto state of a single photon in two modes (e.g. spatial or polarization). We suggest schemes allowing direct probabilistic realization of the fundamental Toffoli and Fredkin gates without resorting to a sequence of single- and two-qubit gates. This yields more compact schemes and potentially reduces the number of ancilla photons. The proposed setups involve passive linear optics, sources of auxiliary single photons or maximally entangled pairs of photons, and single-photon detectors. In parti…
We propose and analyze a probabilistic scheme to entangle two spatially separated topological qubits in a $p_{x}+ip_{y}$ superfluid using controlled collisions between atoms in movable dipole traps and unpaired atoms inside vortex cores in the superfluid. We discuss how to test the violation of Bell's inequality with the generated entanglement. A set of universal quantum gates is shown to be implementable \textit{deterministically} using the entanglement despite the fact that the entangled states can only be created probabilistically.
We propose a probabilistic quantum cloning scheme using Greenberger-Horne-Zeilinger states, Bell basis measurements, single-qubit unitary operations and generalized measurements, all of which are within the reach of current technology. Compared to another possible scheme via Tele-CNOT gate [D. Gottesman and I. L. Chuang, Nature 402, 390 (1999)], the present scheme may be used in experiment to clone the states of one particle to those of two different particles with higher probability and less GHZ resources.
In the framework of quantum computation with mixed states, a fuzzy representation of CNOT gate is introduced. In this representation, the incidence of non-factorizability is specially investigated.
We propose and study a method for using non-maximally entangled states to implement probabilistically non-local gates. Unlike distillation-based protocols, this method does not generate a maximally entangled state at intermediate stages of the process. As a consequences, the method becomes more efficient at a certain range of parameters. Gates of the form $\exp[iξσ_{n_A}σ_{n_B}]$ with $ξ\ll1$, can be implemented with nearly unit probability and with vanishingly small entanglement, while for the distillation-based method the gate is produced with a vanishing success probability. We also derive …
We prove the existence of a class of two--input, two--output gates any one of which is universal for quantum computation. This is done by explicitly constructing the three--bit gate introduced by Deutsch [Proc.~R.~Soc.~London.~A {\bf 425}, 73 (1989)] as a network consisting of replicas of a single two--bit gate.
In this paper, elementary quantum gate operations, such as the phase gate, the controlled-NOT gate, the swap and the Fredkin gate are constructed using joint measurement and pairs of entangled qubit pairs. The relation between the state of the entangled pair and the joint measurement basis is discussed. Some other generalization is also discussed.
We develop a computation model for solving Boolean networks that implements wires through quantum ground-state computation and implements gates through identities following from angular momentum algebra and statistics. The gates are static in the sense that they contribute Hamiltonian 0 and hold as constants of the motion; only the wires are dynamic. Just as a spin 1/2 makes an ideal 1-bit memory element, a spin 1 makes an ideal 3-bit gate. Such gates cost no computation time: relaxing the wires alone solves the network. We compare computation time with that of an easier Boolean network where …
We propose to design multispin quantum gates in which the input and output two-state systems (spins) are not necessarily identical. We describe the motivations for such studies and then derive an explicit general two-spin interaction Hamiltonian which accomplishes the quantum XOR gate function for a system of three spins: two input and one output.
The basic one-bit gates (X, Y, Z, Hadamard, phase, pi/8) as well as the controlled cnot and Toffoli gates are reformulated in the language of geometric-algebra quantum-like computation. Thus, all the quantum algorithms can be reformulated in purely geometric terms without any need of tensor products.
We offer an alternative to the conventional network formulation of quantum computing. We advance the analog approach to quantum logic gate/circuit construction. As an illustration, we consider the spatially extended NOT gate as the first step in the development of this approach. We derive an explicit form of the interaction Hamiltonian corresponding to this gate and analyze its properties. We also discuss general extensions to the case of certain time-dependent interactions which may be useful for practical realization of quantum logic gates.
Composite pulses provide a simple means for constructing quantum logic gates which are robust to small errors in the control fields used to implement them. Here I describe how antisymmetric composite NOT gates can be nested to produce gates with arbitrary tolerance of errors.
I describe the use of techniques based on composite rotations to combat systematic errors in controlled phase gates, which form the basis of two qubit quantum logic gates. Although developed and described within the context of Nuclear Magnetic Resonanace (NMR) quantum computing these sequences should be applicable to any implementation of quantum computation based on Ising couplings. In combination with existing single qubit gates this provides a universal set of robust quantum logic gates.
