Multi-walker discrete time quantum walks on arbitrary graphs, their properties and their photonic implementation

Probably not a substantial claim: may not pass S3 (load-bearing).

At each step the reach of the position state expands outwards across the line.

“At each step the reach of the position state expands outwards across the line.”

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In the case of distinguishable walkers, the coincidence matrix is uncorrelated and can be obtained by multiplying the single photon statistics along one axis by the single photon statistics along the other axis.

“Note that in the case of distinguishable walkers the coincidence matrix is uncorrelated and can be obtained by multiplying the single photon statistics along one axis by the single photon statistics along the other axis.”

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For the 7-level two-photon quantum walk, the projection onto single click events for distinguishable and indistinguishable walkers shows no discernible visual difference, with a small numerical L1 difference of 0.04.

“Visually there is no discernable difference in (c) between the distinguishable probabilities ( P (1) i ) and indistinguishable probabilities ( P (2) i ), although there is a small numerical difference L 1 = 1 2 ∑ i | P (1) i -P (2) i | = 0 . 04.”

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For indistinguishable walkers in the 4-level walk, H(P) is necessarily greater than zero, and in this example H(P)=1.74.

“On the other hand, for indistinguishable walkers H ( P ) is necessarily greater than zero. In this example we have H ( P ) = 1 . 74, a characteristic of photon correlations.”

Classified as probably not a substantial claim.

For a single beamsplitter with two inputs and two outputs, putting a single photon into each port transforms the state |11> to (|20> + |02>)/√2.

“Should we put a single photon into each beamsplitter port the system undergoes the transformation | 11 〉 → ( | 20 〉 + | 02 〉 ) / √ 2.”

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Upon number-resolving detection after that transformation, both photons are always measured at one port or both at the other, each with 50% probability, and one photon at each output is never measured.

“Thus upon performing number resolving detection, we will always measure both photons at one port or both photons at the other (each with 50% probability), whereas one photon at each output will never be measured.”

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In the JPD of indistinguishable particles, clear oscillations exist between different terms, distinct from the distinguishable case, and this structure is lost upon projection into one dimension.

“Notice that in the JPD of indistinguishable particles exist clear oscillations between different terms, which are distinct from the distinguishable case. This structure is lost upon projection into one dimension.”

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The first experiments investigating the impact of correlations in quantum walks with two walkers were done in the continuous time model, showing correlations by classical intensity correlation measurements and using coincidence measurements of two photons of a PDC source.

“The first experiments investigating the impact of correlations in quantum walks with two walkers were done in the continuous time model: Bromberg et. al [15] showed correlations by classical intensity correlation measurements and Peruzzo et. al [16] used coincidence measurements of two photons of a PDC source, both with integrated wave guide devices.”

• cites [15] sourceNot checked yet

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• cites [16] sourceNot checked yet

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Quantum walks involving more particles unlock additional possibilities unavailable in classical random walks, as quantum particles can be entangled or indistinguishable fermions or bosons.

“Indeed, quantum walks involving more particles unlock additional possibilities which are unavailable in classical random walks, as the quantum particles can be entangled or they can be indistinguishable fermions or bosons.”

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Quantum walks offer an alternate approach to implementing quantum algorithms compared to the typical circuit or cluster state models for quantum algorithms.

“Quantum walks [1] offer an alternate approach to implementing quantum algorithms [2] compared to the typical circuit [3] or cluster state [4, 5] models for quantum algorithms.”

• cites [1] sourceNot checked yet

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• cites [2] Aharonov D Ambainis A Kempe J Vazirani U, 2001Not checked yet

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• cites [3] Nielsen M A, 2000, Quantum Computation and Quantum InformationNot checked yet

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Using the formalism, the walk is characterized completely by the initial state and the unitary coin matrices of the graph; the evolution of a particular walker is characterized entirely by its neighborhood.

“Using this formalism the walk is characterized completely by (1) the initial state of the system, and (2) the unitary coin matrices of the graph. Additionally, the evolution of a particular walker is characterized entirely by its neighborhood in the graph.”

• AssumptionThe authors state this as an assumption: The coin is independent of the number of walkers (no nonlinear cross-Kerr effects), so propagation is defined entirely by linear elements.

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The algorithm implemented by a quantum walk is characterized entirely by the choice of coin and step operators.

“In fact, the algorithm implemented by a quantum walk is characterized entirely by this choice.”

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Quantum walks have been successfully applied to database search and graph isomorphism problems.

“Indeed, quantum walks have been successfully applied to database search [6] and graph isomorphism [7] problems.”

• cites [6] sourceNot checked yet

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• cites [7] Douglas B L, 2008, J. Phys. A: Math Theor. sourceNot checked yet

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For arbitrary graphs with varying degree, a single coin cannot describe the dynamics of the entire graph; unique A(x) operators must be defined for every neighborhood.

“In general, arbitrary graphs will have varying degree, i.e. different neighborhoods contain different numbers of vertices. Thus in general we cannot use a single coin to describe the dynamics of the entire graph. Instead, unique A ( x ) operators must be defined for every neighborhood.”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

Photonic discrete time quantum walks were experimentally demonstrated using weak coherent light and using single photons produced via parametric down-conversion.

“Indeed, photonic discrete time quantum walks were experimentally demonstrated by Schreiber et al. [13] using weak coherent light and by Broome et. al [14] using single photons produced via parametric down-conversion (PDC).”

• cites [13] sourceNot checked yet

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• cites [14] sourceNot checked yet

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Probably not a substantial claim: may not pass S3 (load-bearing).

The choice of coin and step operators can vary at each step of the walk.

“Not only can we choose arbitrary operators, but, in general, the choice can vary at each step of the walk.”

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The universality of quantum walks has been shown for continuous and discrete systems.

“More recently, the universality of quantum walks has been shown in the case of continuous [8] and discrete [9] systems.”

• cites [8] sourceNot checked yet

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• cites [9] sourceNot checked yet

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The step operator in the general formalism must change both the coin and the position to ensure unitarity; the analogous transformation that updates only position would not be reversible.

“This slight change in the formalism is necessary to ensure unitarity of the step operator. For example, if in analogy to Eq. 1 our step operator implemented the transformation w ( x, j ) † → w ( j, j ) † , the operation would not be reversible.”

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In general, coherent states can only simulate multiple walkers in special cases: with distinguishable photons, which generate separable statistics, and when multiple walkers begin in the same initial state.

“Thus, in general, coherent states can only simulate multiple walkers in special cases: (1) with distinguishable photons, which generate separable statistics, and (2) the example presented above where multiple walkers begin in the same initial state.”

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In an arbitrary quantum walk, distinguishable walkers are possible with an appropriate choice of coin and step operators, and this arises naturally with distinguishable photons.

“However, in an arbitrary quantum walk, distinguishable walkers are also possible with an appropriate choice of coin and step operators. This arises naturally with distinguishable photons.”

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For distinguishable walkers, the coincidence probability contains no interference terms and has an uncorrelated form, i.e., it can be represented as the product of two independent functions.

“This expression contains no interference terms, as expected. It is evident from Eq. 11 that for distinguishable walkers the coincidence probability has an uncorrelated form, i.e. it can be represented as the product of two independent functions.”

Probably not a substantial claim: may not pass S3 (load-bearing).

For indistinguishable photons, normalization of the JPD requires the sum of the lower triangular elements of the matrix to be 1.

“In the case of indistinguishable photons, for normalization we require that the sum of the lower triangular elements of the matrix be 1, ∑ N i =1 ∑ i j =1 P ij = 1, where N is the number of output modes.”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

Because the position states propagate in both directions, interference between the position states takes place; this is the defining feature that differentiates the quantum walk from a classical random walk.

“Additionally, because the position states propagate in both directions, interference between the position states takes place. This is the defining feature that differentiates the quantum walk from a classical random walk [27].”

• cites [27]Not checked yet

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Probably not a substantial claim: may not pass S3 (load-bearing).

The JPD can be used to derive any output statistics of interest.

“The JPD can be used to derive any output statistics of interest.”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

The particular choice of coin and step operators for the one-dimensional walk can be optically implemented using 50/50 beamsplitters.

“This particular choice of coin and step operators can be optically implemented using 50/50 beamsplitters.”

• AssumptionThe authors state this as an assumption: The coin is independent of the number of walkers (no nonlinear cross-Kerr effects), so propagation is defined entirely by linear elements.

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

In general, time-dependent coins are allowed, and the evolution of the system is of the form ∏ i SC_i.

“In general, time-dependent coins are allowed, and the evolution of the system will be of the form ∏ i SC i , where C i is the coin operator at time step i .”

May combine two claims.

When incident photons are in the same spatio-temporal mode, they will interfere within the quantum walk; when distinguishable, they will not interfere and evolve independently.

“In the former case the incident photons will interfere within the quantum walk, whereas in the later case they will not and instead evolve completely independently of one another and no interference will be observed.”

May combine two claims.

Differences in the meeting probability can be found also in the asymptotic regime, but quantum effects on the long-time scale are much more subtle, as discussed in Ref. [20].

“Moreover, the differences in the meeting probability can be found also in the asymptotic regime. However, the quantum effects on the long-time scale are much more subtle, as discussed in detail in Ref. [20].”

• cites [20] Štefaňák M, 2006, J. Phys. A: Math. Gen. sourceNot checked yet

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Fully characterizing the statistics of a multi-walker quantum walk is infeasible with large numbers of walkers, necessitating development of algorithm-specific metrics.

“This implies that fully characterizing the statistics of a multi-walker quantum walk is infeasible with large numbers of walkers, necessitating the development of algorithm specific metrics for characterizing the system which do not require exponentially large numbers of measurements.”

Probably not a substantial claim: may not pass S4 (based). May combine two claims.

Indistinguishable walkers interfere with one another, and this is the interesting case.

“In the following sections we will primarily consider the situation of indistinguishable walkers, since this is the interesting case where the walkers interfere with one another.”

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Peruzzo et al. experimentally demonstrated an optical implementation of a two photon quantum walk which overlaps with the techniques presented here.

“While preparing this manuscript Peruzzo et. al [16] experimentally demonstrated an optical implementation of a two photon quantum walk which overlaps with the techniques presented here.”

• cites [16] sourceNot checked yet

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Upon measurement of a single walker, the factorizable nature of the two-walker state is lost, and the resulting state contains cross-combinations from different terms.

“Thus, upon measurement the factorizeable nature of the state is lost, and our resulting state contains crosscombinations from the different terms in the initially factorizeable state.”

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In the case of distinguishable walkers, {m,n} and {n,m} are distinct, leading to a different normalization condition than for indistinguishable walkers.

“In the case of distinguishable walkers, { m,n } and { n, m } are now distinct. This leads to a different normalization condition than for the indistinguishable case [32].”

• cites [32]Not checked yet

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• AssumptionThe authors state this as an assumption: Detectors can resolve the distinct modes a and b.

Probably not a substantial claim: may not pass S3 (load-bearing).

This correlation was experimentally observed by Bromberg et al. in a continuous time quantum walk.

“This was experimentally observed by Bromberg et. al in a continuous time quantum walk [15].”

• cites [15] sourceNot checked yet

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The two-photon JPD and its projection onto one dimension can both be directly experimentally measured.

“The two photon JPD and its projection onto one dimension can both be directly experimentally measured.”