Showing papers in "Quantum Information & Computation in 2006"
TL;DR: A new version of the quantum threshold theorem is proved that applies to concatenation of a quantum code that corrects only one error, and this theorem is used to derive arigorous lower bound on the quantum accuracy threshold e0, the best lower bound that has been rigorously proven so far.
Abstract: We prove a new version of the quantum threshold theorem that applies to concatenationof a quantum code that corrects only one error, and we use this theorem to derive arigorous lower bound on the quantum accuracy" threshold e0. Our proof also appliesto concatenation of higher-distance codes, and to noise models that allow faults to becorrelated in space and in time. The proof uses new criteria for assessing the accuracy" offault-tolerant circuits, which are particularly conducive to the inductive analysis of recur-sire simulations. Our lower bound on the threshold, e0 ≥ 2.73 × 10-5 for an adversarialindependent stochastic noise model, is derived from a computer-assisted combinatorialanaly sis; it is the best lower bound that has been rigorously proven so far.
440 citations
TL;DR: The algorithm can be used to compile Shor's algorithm into an efficient fault-tolerant form using only Hadamard, controlled-not, and π/8 gates, and is generalized to apply to multi-qubit gates and togates from SU(d).
Abstract: This pedagogical review presents the proof of the Solovay-Kitaev theorem in the form ofan efficient classical algorithm for compiling an arbitrary single-qubit gate into a sequenceof gates from a fixed and finite set. The algorithm can be used, for example, to compileShor's algorithm, which uses rotations of π/2k, into an efficient fault-tolerant form usingonly Hadamard, controlled-not, and π/8 gates. The algorithm runs in O(log2.71(1/e))time, and produces as output a sequence of O(log3.97(1/e)) quantum gates which isguaranteed to approximate the desired quantum gate to an accuracy within e > 0. Wealso explain how the algorithm can be generalized to apply to multi-qubit gates and togates from SU(d).
317 citations
TL;DR: This work reduces the cost of addition dramatically with only a slight increase in the number of required qubits, and can be used within current modularmultiplication circuits to reduce substantially the run-time of Shor's algorithm.
Abstract: We present an efficient addition circuit, borrowing techniques from classical carry-lookahead arithmetic. Our quantum carry-lookahead (QCLA) adder accepts two n-bitnumbers and adds them in O(log n) depth using O(n) ancillary qubits. We present bothin-place and out-of-place versions, as well as versions that add modulo 2n and modulo2n - 1. Previously, the linear-depth ripple-carry addition circuit has been the methodof choice. Our work reduces the cost of addition dramatically with only a slight increasein the number of required qubits. The QCLA adder can be used within current modularmultiplication circuits to reduce substantially the run-time of Shor's algorithm.
241 citations
TL;DR: In this article, the Operator Quantum Error Correction formalism was introduced, which is a new scheme for the error correction of quantum operations that incorporates the known techniques, i.e., the standard error correction model, the method of decoherence-free subspaces, and the noiseless subsystem method, as special cases.
Abstract: This paper is an expanded and more detailed version of the work [1] in which the Operator Quantum Error Correction formalism was introduced. This is a new scheme for the error correction of quantum operations that incorporates the known techniques -- i.e. the standard error correction model, the method of decoherence-free subspaces, and the noiseless subsystem method -- as special cases, and relies on a generalized mathematical framework for noiseless subsystems that applies to arbitrary quantum operations. We also discuss a number of examples and introduce the notion of "unitarily noiseless subsystems".
150 citations
TL;DR: Several Finsler metrics whose minimal length geodesics provide lower bounds on quantum circuitsize are constructed and a large class of solutions to the geodesic equation, which are called Pauli geodesICS, since they arise from isometries generated by the Pauli group.
Abstract: What is the minimal size quantum circuit required to exactly implement a specified n-qubit unitary operation, U, without the use of ancilla qubits? We sbow that a lowerbound on the minimal size is provided by the length of the minimal geodesic between Uand the identity, I, where length is defined by a suitable Finsler metric on the manifoldSU(2n). The geodesic curves on these manifolds have the striking property that oncean initial position and velocity are set, the remMnder of the geodesic is completelydeternfined by a second order differential equation known as the geodesic equation. Thisis in contrast with the usual case in circuit design, either classical or quantum, wherebeing given part of an optimal circuit does not obviously assist in the design of therest of the circuit. Geodesic analysis thus offers a potentially powerful approacb to theproblem of proving quantum circuit lower bounds. In this paper we construct severalFinsler metrics whose minimal length geodesics provide lower bounds on quantum circuitsize. For eacb Finsler metric we give a procedure to compute the corresponding geodesicequation. We also construct a large class of solutions to the geodesic equation, whichwe call Pauli geodesics, since they arise from isometries generated by the Pauli group.For any unitary U diagonal in the computational basis, we sbow that: (a) proposed theminimal length geodesic is unique, it must be a Pauli geodesic; (b) finding the length ofthe minimal Pauli geodesic passing from I to U is equivalent to solving an exponentialsize instance of the closest vector in a lattice problem (CVP); and (c) all but a doublyexponentially small fraction of sucb unit aries have nfinimal Pauli geodesics of exponentiallength.
141 citations
TL;DR: In this article, it was shown that the W-state is the only pure state that can be used to exactly solve the problem of leader election in anonymous quantum networks, while the GHZ-state cannot be used for distributed consensus when no classical post-processing is considered.
Abstract: It is well understood that the use of quantum entanglement significantly enhances the computational power of systems. Much of the attention has focused on Bell states and their multipartite generalizations. However, in the multipartite case it is known that there are several inequivalent classes of states, such as those represented by the W-state and the GHZ-state. Our main contribution is a demonstration of the special computational power of these states in the context of paradigmatic problems from classical distributed computing. Concretely, we show that the W-state is the only pure state that can be used to exactly solve the problem of leader election in anonymous quantum networks. Similarly we show that the GHZ-state is the only one that can be used to solve the problem of distributed consensus when no classical post-processing is considered. These results generalize to a family of W- and GHZ-like states. At the heart of the proofs of these impossibility results lie symmetry arguments.
83 citations
TL;DR: Improvements in readout speed and fidelity are discussed in the context of scalable quantum computation architectures.
Abstract: Simultaneous measurement of multiple qubits stored in hyperfine levels of trapped 111Cd+ ions is realized with an intensified charge-coupled device (CCD) imager. A general theory of fluorescence detection for hyperfine qubits is presented and applied to experimental data. The use of an imager for photon detection allows for multiple qubit state measurement with detection fidelities of greater than 98%. Improvements in readout speed and fidelity are discussed in the context of scalable quantum computation architectures.
71 citations
TL;DR: Analytical results showing that decoherence can be useful for mixing time in a continuous-time quantum walk on finite cycles are proved and the existence of a unique optimal rate for which the mixing time is minimized is confirmed.
Abstract: We prove analytical results showing that decoherence can be useful for mixing time in a continuous-time quantum walk on finite cycles. This complements the numerical observations by Kendon and Tregenna (Physical Review A 67 (2003), 042315) of a similar phenomenon for discrete-time quantum walks. Our analygicM treatment of continuous-time quantum walks includes a continuous monitoring of all vertices that induces the decoherence process. We identify the dymamics of the probability distribution and observe how mixing times undergo the transition from quantum to classical behavior as our decoherence parameter grows from zero to infinity. Our results show that, for small rates of decoherence, the mixing time improves linearly with decoherence, whereas for large rates of decoherence, the mixing time deteriorates linearly towards the classical limit. In the middle region of decoherence rates, our numerical data confirnm the existence of a unique optimal rate for which the mixing time is minimized.
48 citations
TL;DR: In this article, the authors define pseudothresholds and present classical and quantum fault-tolerant circuits exhibiting pseudo-thresholds that differ by a factor of 4 from faulttolerance thresholds for typical relationships between component failure rates.
Abstract: An arbitrarily reliable quantum computer can be efficiently constructed from noisy components using a recursive simulation procedure, provided that those components fail with probability less than the fault-tolerance threshold. Recent estimates of the threshold are near some experimentally achieved gate ?delities. However, the landscape of threshold estimates includes pseudothresholds, threshold estimates based on a subset of components and a low level of the recursion. In this paper, we observe that pseudothresholds are a generic phenomenon in fault-tolerant computation. We define pseudothresholds and present classical and quantum fault-tolerant circuits exhibiting pseudothresholds that differ by a factor of 4 from fault-tolerance thresholds for typical relationships between component failure rates. We develop tools for visualizing how reliability is influenced by recursive simulation in order to determine the asymptotic threshold. Finally, we conjecture that refinements of these methods may establish upper bounds on the fault-tolerance threshold for particular codes and noise models.
42 citations
TL;DR: It is shown that it is desirable for the electrode structure to produce a d.c. octupole moment with ana.
Abstract: We study the problem of designing electrode structures that allow pairs of ions to bebrought together and separated rapidly in an array of linear Paul traps. We show thatit is desirable for the electrode structure to produce a d.c. octupole moment with ana.c. radial quadrupole. For the case where electrical breakdown limits the voltagesthat can be applied, we show that the octupole is more demanding than the quadrupolewhen the characteristic distance scale of the structure is larger than 1 to 10 microns (fortypical materials). We present a variety of approaches and optimizations of structuresconsisting of one to three layers of electrodes. The three-layer structures allow the fastestoperation at given distance ρ from the trap centres to the nearest electrode surface, butwhen the total thickness w of the structure is constrained, leading to w < ρ, then twolayerstructures may be preferable.
40 citations
TL;DR: A quantum circuit for Shor's factoring algorithm that uses 2n + 2 qubits, where n is the length of the number to be factored and O(n3 log n) is the depth and size.
Abstract: We construct a quantum circuit for Shor's factoring algorithm that uses 2n + 2 qubits,where n is the length of the number to be factored. The depth and size of the circuitare O(n3) and O(n3 log n), respectivel). The number of qubits used in the circuit is lessthan that in any other quantum circuit ever constructed for Shor's factoring algorithm.Moreover, the size of the circuit is about half the size of Beauregard's quantum circuitfor Shor's factoring algorithm, which uses 2n + 3 qubits.
TL;DR: Numerical simulation is used to investigate how entanglement between register qubits varies as Shor's algorithm is run on a quantum computer, and the shifting patterns in theEntanglement are found to relate to the choice of basis for the quantum Fourier transform.
Abstract: Entanglement has been termed a critical resource for quantum information processing and is thought to be the reason that certain quantum algorithms, such as Shor's factoring algorithm, can achieve exponentially better performance than their classical counterparts The nature of this resource is still not fully understood: here we use numerical simulation to investigate how entanglement between register qubits varies as Shor's algorithm is run on a quantum computer The shifting patterns in the entanglement are found to relate to the choice of basis for the quantum Fourier transform
TL;DR: In this article, a new algorithm that translates a unitary matrix into a quantum circuit according to the G = KAK theorem in Lie group theory is presented, which can automatically reproduce the well-known efficient quantum circuit for the n-qubit quantum Fourier transform.
Abstract: We provide a new algorithm that translates a unitary matrix into a quantum circuit accordingto the G = KAK theorem in Lie group theory. With our algorithm, any matrixdecomposition corresponding to type-AIII KAK decompositions can be derived according to thegiven Cartan involution. Our algorithm contains, as its special cases, CosineSine decomposition(CSD) and Khaneja-Glaser decomposition (KGD) in the sense thatit derives the same quantum circuits as the ones obtained by them if we select suitableCartan involutions and square root matrices. The selections of Cartan involutions forcomputing CSD and KGD will be shown explicitly. As an example, we show explicitlythat our method can automatically reproduce the well-known efficient quantum circuitfor the n-qubit quantum Fourier transform.
TL;DR: A different algorithm is presented here, which has the same worst-case behavior as the Phase-π/3 search algorithm but much better average- case behavior and is an example of how measurement can allow us to bypass some restrictions imposed by unitarity on quantum computing.
Abstract: The standard quantum search lacks a feature, enjoyed by many classical algorithms, of having a fixed point, i.e. monotonic convergence towards the solution. Recently a fixed point quantum search algorithm has been discovered, referred to as the Phase-π/3 search algorithm, which gets around this limitation. While searching a database for a target state, this algorithm reduces the error probability from e to e2q+1 using q oracle queries, which has since been proved to be asymptotically optimal. A different algorithm is presented here, which has the same worst-case behavior as the Phase-π/3 search algorithm but much better average-case behavior. Furthermore the new algorithm gives e2q+1 convergence for all integral q, whereas the Phase-π/3 search algorithm requires q to be (3n -1)/2 with n a positive integer. In the new algorithm, the operations are controlled by two ancilla qubits, and fixed point behavior is achieved by irreversible measurement operations applied to these ancillas. It is an example of how measurement can allow us to bypass some restrictions imposed by unitarity on quantum computing.
TL;DR: This work considers the problem of remote state preparation, in the presence of entanglement and in the scenario of single use of the channel, and studies the communication complexity of this problem.
Abstract: We consider the problem of remote state preparation recently studied in several papers. We study the communication complexity of this problem, in the presence of entanglement and in the scenario of single use of the channel.
TL;DR: In this paper, the advantage of pure-state quantum computation without entanglement over classical computation was studied, and it was shown that the Deutsch-Jozsa algorithm still has an advantage over the classical ones.
Abstract: We study the advantage of pure-state quantum computation without entanglement over classical computation. For the Deutsch-Jozsa algorithm we present the maximal subproblem that can be solved without entanglement, and show that the algorithm still has an advantage over the classical ones. We further show that this subproblem is of greater significance, by proving that it contains all the Boolean functions whose quantum phase-oracle is non-entangling. For Simon's and Grover's algorithms we provide simple proofs that no non-trivial subproblems can be solved by these algorithms without entanglement.
TL;DR: This work completes the characterization of the quantum Fisher information metrics by providing a closed andractable formula for the set of Morozova-Chentsov functions and provides a continuously increasing bridge between the smallest and largest symmetric monotonemetrics.
Abstract: The quantum Fisher information is a Riemannian metric, defined on the state space ofa quantum system, which is symmetric and decreasing under stochastic mappings Contraryto the classical case such a metric is not unique We complete the characterization,initiated by Morozova, Chentsov and Petz, of these metrics by providing a closed andtractable formula for the set of Morozova-Chentsov functions In addition, we providea continuously increasing bridge between the smallest and largest symmetric monotonemetrics
TL;DR: In this article, it was shown that the gate library containing arbitrary local unitaries and one two-qudit gate, CINC, is exact universal and can be implemented without ancillas without changing the asymptotics.
Abstract: This paper concerns the efficient implementation of quantum circuits for qudits. We show that controlled two-qudit gates can be implemented without ancillas and prove that the gate library containing arbitrary local unitaries and one two-qudit gate, CINC, is exact-universal. A recent paper [S.Bullock, D.O'Leary, and G.K. Brennen, Phys. Rev. Lett. 94, 230502 (2005)] describes quantum circuits for qudits which require O(dn) two-qudit gates for state synthesis and O(dn2) two-qudit gates for unitary synthesis, matching the respective lower bound complexities. In this work, we present the state-synthesis circuit in much greater detail and prove that it is correct. Also, the ⌈(n-2)/(d-2)⌉ ancillas required in the original algorithm may be removed without changing the asymptotics. Further, we present a new algorithm for unitary synthesis, inspired by the QR matrix decomposition, which is also asymptotically optimal.
TL;DR: It is shown that, regardless of the number of ancillae arbitrary arity Toffoli gates cannot besimulated exactly by a constant depth circuit family with gates of bounded arity, and that parity requires log depth quantum circuits.
Abstract: We consider the resource bounded quantum circuit model with circuits restricted by thenumber of qubits they act upon and by their depth. Focusing on natural universal setsof gates which are familiar from classical circuit theory, several new lower bounds forconstant depth quantum circuits are proved. The main result is that parity (and hencefanout) requires log depth quantum circuits, when the circuits are composed of singlequbit and arbitrary size Toffoli gates, and when they use only constantly many ancillae.Under this constraint, this bound is close to optimal. In the case of a non-constantnumber a of ancillae and n input qubits, we give a tradeoff between a and the requireddepth, that results in a non-constant lower bound for fanout when a = n1-o(1). We alsoshow that, regardless of the number of ancillae arbitrary arity Toffoli gates cannot besimulated exactly by a constant depth circuit family with gates of bounded arity.
TL;DR: This paper demonstrates an implementation of quantum key distribution with continuous variables based on a go-&-return configuration over distances up to 14km that leads to self-compensation of polarisation and phase fluctuations.
Abstract: We demonstrate an implementation of quantum key distribution with continuous variables based on a go-&-return configuration over distances up to 14km. This configuration leads to self-compensation of polarisation and phase fluctuations. We observe a high degree of stability of our set-up over many hours.
TL;DR: The distillability problem can be reformulated as a special instance of the separability problem, for which a large number of tools and techniques are available, and the results suggest that bound entanglement is primarily a phenomenon found in low dimensional quantum systems.
Abstract: An important open problem in quantum information theory is the question of the existence of NPT bound entanglement. In the past years, little progress has been made, mainly because of the lack of mathematical tools to address the problem. (i) In an attempt to overcome this, we show how the distillability problem can be reformulated as a special instance of the separability problem, for which a large number of tools and techniques are available. (ii) Building up to this we also show how the problem can be formulated as a Schmidt number problem. (iii) A numerical method for detecting distillability is presented and strong evidence is given that all 1-copy undistillable Werner states are also 4-copy undistillable. (iv) The same method is used to estimate the volume of distillable states, and the results suggest that bound entanglement is primarily a phenomenon found in low dimensional quantum systems. (v) Finally, a set of one parameter states is presented which we conjecture to exhibit all forms of distillability.
TL;DR: It is proved that there are exactly 6 independent criteria for three parties and 22 for four parties and it is conjecture that the remaining class of criteria only contains truly independent permutation separability criteria.
Abstract: Recently, P Wocjan and M Horodecki [Open Syst Inf Dyn 12, 331 (2005)] gave a characterizationof combinatorially independent permutation separability criteria Combinatorialindependence is a necessary condition for permutations to yield truly independentcriteria meaning that no criterion is strictly stronger that any other In this paper weobserve that some of these criteria are still dependent and analyze why these dependenciesoccur To remove them we introduce an improved necessary condition and give acomplete classification of the remaining permutations We conjecture that the remainingclass of criteria only contains truly independent permutation separability criteria Ourconjecture is based on the proof that for two, three and four parties all these criteria aretruly independent and on numerical verification of their independence for up to 8 partiesIt was commonly believed that for three parties there were 9 independent criteria, herewe prove that there are exactly 6 independent criteria for three parties and 22 for fourparties
TL;DR: A description of relevant results in standard cryptography and in the design of αη to put the above issues in the proper framework and to elucidate some security features of this new approach to quantum cryptography.
Abstract: Lo and Ko in [1] have developed some attacks on the cryptosystem called αη [2], claimingthat these attacks undermine the security of αη for both direct encryption and keygeneration. In this paper, we show that their arguments fail in many different ways.In particular, the first attack in [1] requires channel loss or length of known-plaintextthat is exponential in the key length and is unrealistic even for moderate key lengths.The second attack is a Grover search attack based on 'asymptotic orthogonality' andwas not analyzed quantitatively in [1]. We explain why it is not logically possible to"pull back" an argument valid only at n = ∞ into a limit statement, let alone one validfor a finite number of transmissions n. We illustrate this by a 'proof' using a similarasymptotic orthogonality argument that coherent-state BB84 is insecure for any value ofloss. Even if a limit statement is true, this attack is a priori irrelevant as it requires anindefinitely large amount of known-plaintext, resources and processing. We also explainwhy the attacks in [1] on αη as a key-generation system are based on misinterpretations of[2]. Some misunderstandings in [1] regarding certain issues in cryptography and opticalcommunications are also pointed out. Short of providing a security proof for αη, weprovide a description of relevant results in standard cryptography and in the designof αη to put the above issues in the proper framework and to elucidate some securityfeatures of this new approach to quantum cryptography.
TL;DR: A novel deterministic quantum key distribution protocol based on Bell-state measurements of hyperentangled photon states and a scheme for a probabilistic controlled-not gate which operates with a 50 % success probability are proposed.
Abstract: We discuss quantum information processing with hyperentangled photon states - statesentangled in multiple degrees of freedom. Using an additional entangled degree of freedomas an ancilla space, it has been shown that it is possible to perform efficient Bell-state measurements. We briefly review these results and present a novel deterministicquantum key distribution protocol based on Bell-state measurements of hyperentangledphotons. In addition, we propose a scheme for a probabilistic controlled-not gate whichoperates with a 50 % success probability. We also show that despite its probabilisticnature, the controlled-not gate can be used for an efficient, nonlocal demonstration ofthe Deutsch algorithm using two separate photons.
TL;DR: In this paper, the stability of Shor's factorization algorithm when exposed to a discrete error model was investigated and it was shown that the error locations within the circuit itself heavily influences the probability of success of the QPF subroutine.
Abstract: Shor's factorisation algorithm is a combination of classical pre- and post-processing and a quantum period finding (QPF) subroutine which allows an exponential speed up over classical factoring algorithms. We consider the stability of this subroutine when exposed to a discrete error model that acts to perturb the computational trajectory of a quantum computer. Through detailed state vector simulations of an appropriate quantum circuit, we show that the error locations within the circuit itself heavily influences the probability of success of the QPF subroutine. The results also indicate that the naive estimate of required component precision is too conservative.
TL;DR: A compact and exact expression for the probability distribution of the entanglement values across any bipartite cut is obtained, which allows for exact derivations of the average entanglements and the degree of concentration of measure around this average.
Abstract: We study the entanglement properties of random pure stabilizer states in spin- 1/2 particles. We obtain a compact and exact expression for the probability distribution of the entanglement values across any bipartite cut. This allows for exact derivations of the average entanglement and the degree of concentration of measure around this average. We also give simple bounds on these quantities. We find that for large systems the average entanglement is near maximal and the measure is concentrated around it.
TL;DR: This paper gives new observations on the mixing dynamics of a continuous-time quantumwalk on circulants and their bunkbed extensions through the join G + H and the Cartesian product G × H of graphs G and H.
Abstract: This paper give new observations on the mixing dynamics of a continuous-time quantumwalk on circulants and their bunkbed extensions These bunkbeds are defined throughtwo standard graph operators: the join G + H and the Cartesian product G × H ofgraphs G and H Our results include the following: (i) The quantum walk is average uniform mixing oncirculants with bounded eigenvalue multiplicity This extends a known fact about thecycles Cn (ii) Explicit analysis of the probability distribution of the quantum walk onthe join of circulants This explains why complete partite graphs are not averageuniform mixing, using the fact Kn = K1 + Kn-1 and Kn,n = Kn + + Kn(iii)The quantum walk on the Cartesian product of a m-vertex path Pm and a circulantG, namely, Pm × G, is average uniform mixing if G is This highlights a differencebetween circulants and the hypercubes Qn = P2 × Qn-1Our proofs employ purely elementary arguments based on the spectra of the graphs
TL;DR: This approach based on channels gives rise to a unified picture of known and new bounds on the classical information (Holevo's, Shumacher-Westmoreland-Wootters', Hall's, Scutaru's bounds, anew upper bound and a new lower one). Some examples clarify the mutual relationships among the various bounds.
Abstract: While a positive operator valued measure gives the probabilities in a quantum measurement,an instrument gives both the probabilities and the a posteriori states. Byinterpreting the instrument as a quantum channel and by using the monotonicity theoremfor relative entropies many bounds on the classical information extracted in aquantum measurement are obtained in a unified manner. In particular, it is shown thatsuch bounds can all be stated as inequalities between mutual entropies. This approachbased on channels gives rise to a unified picture of known and new bounds on the classicalinformation (Holevo's, Shumacher-Westmoreland-Wootters', Hall's, Scutaru's bounds, anew upper bound and a new lower one). Some examples clarify the mutual relationshipsamong the various bounds.
TL;DR: In this article, a new version of the quantum threshold theorem that applies to concatenation of a quantum code that corrects only one error is presented, and this theorem is used to derive an arigorous lower bound on the q-threshold.
Abstract: We prove a new version of the quantum threshold theorem that applies to concatenationof a quantum code that corrects only one error, and we use this theorem to derive arigorous lower bound on the q...
TL;DR: This work establishes relations between conifold, Segre variety, Hopf fibration, and separablesets of pure two-qubit states and constructs entanglement monotones for multi-qu bit states.
Abstract: We establish relations between conifold, Segre variety, Hopf fibration, and separablesets of pure two-qubit states. Moreover, we investigate the geometry and topology ofseparable sets of pure multi-qubit states based on the complex multi-projective Segrevariety and higher order Hopf fibration. We show that the Segre variety and Hopffibration give a unified geometrical and topological picture of multi-qubit states. Wealso construct entanglement monotones for multi-qubit states.