Decoherence in quantum walks – a review

In the paper

“Both spread linearly, but with different constant prefactors, so both show a quadratic speed up in their spreading compared to classical random walks.”

In the paper

“Using a hopping rate of γ = 0.5 so that the total hopping probability per unit time is one, after 40 units of time we get the probability distribution shown in Figure 2.”

In the paper

“The two types of walks propagate at somewhat different speeds, as evidenced by the different time instants at which they have the same width.”

In the paper

“The continuous-time quantum walk on the line is always symmetric if the hopping rate is the same for both directions”

In the paper

“Equation ( 28) has a straightforward solution in terms of Bessel functions,”

In the paper

“Nonetheless, the continuous-time walk has a very oscillatory nature, especially in the central region.”

In the paper

“where θ measures the strength of the dephasing, with θ = π/4 being complete dephasing (classical walk) and θ = 0 being a pure quantum walk (the approximation above is not valid for exactly θ = 0).”

In the paper

“Decoherence on the coin only does not produce a top-hat distribution, instead the distribution develops a cusp as it passes from quantum to classical.”

In the paper

“The bounding procedure applied here is reasonably accurate -numerical studies give the coefficient of p in the above expansion as 0.09566, compared with the bound of 0.20711.”

In the paper

“The optimum decoherence rate depends on the number of steps in the walk: p u T 2.6 for decoherence on both coin and position and p u T 5 for decoherence on the position only (obtained numerically in Kendon and Tregenna (2003)).”

What the sources say

• Not checked yetcites Kendon and Tregenna (2003)

In the paper

“From the continued expansion of equation ( 72), the prefactor is q 4 (1q) T -4 , so for small q and large T , we have σ 2 (T , q) T (1 + q 4 ), and for the standard deviation σ(T , q) √ T 1 + q 4 2 (77 ) to lowest order in q ≡ 1p.”

In the paper

“The slope of σ(T , p) is finite as p → 0 and zero at p = 1.”

In the paper

“Their work is numerical, but shows that imperfect shifts, or spontaneous tunnelling, smear out the distribution without necessarily reducing the quantum features.”

What the sources say

• Not checked yetcites Dür et al. (2002)

In the paper

“Ryan et al. also applied decoherence artificially to the qubits to observe that the quantum walk degrades into the classical walk.”

What the sources say

• Not checked yetcites Ryan et al. 2005

In the paper

“There are a number of ways in which photons can be made to follow a quantum walk dynamics, and while some have been implemented, all have used high light intensities that admit a classical wave description.”

In the paper

“there are many gaps in the analysis of decoherence in quantum walks, and, more importantly, many gaps in our understanding of what might make a useful quantum walk algorithm.”

In the paper

“For example, no study of decoherence in quantum walks on directed graphs has been done, nor consideration of the class of graphs that might be the pattern of a CP map (rather than of a unitary matrix).”

In the paper

“The continuous-time walk does not produce quite such a nice top-hat as the discrete-time walk, partly due to having support on all sites rather than alternate sites, though it would probably be equally useful in practice.”

In the paper

“Many of the examples studied so far turn out on closer inspection to be exceptional cases, in that a slight change in the symmetry of the dynamics or initial state can reverse the interesting quantum properties.”

In the paper

“In contrast to the extensive studies of the discrete-time walk on the line, there are no published analytical treatments of decoherence in a continuous-time quantum walk on the line.”

In the paper

“there is no reason why p in equations ( 85) and ( 86) cannot be made arbitrarily large.In Figure 11, p = 8000 is shown: the walk hardly manages to leave the starting point.”

In the paper

“The optimal decoherence rate turns out just to remove all the quantum entanglement, and can be thought of as removing the 'extra' quantum correlations that would otherwise keep the distribution away from uniform.”

What the sources say

• Not checked yetcites Kendon and Maloyer (2006)

In the paper

“While this does mix asymptotically, in general, like the discretetime walk, the limiting distribution is not uniform: it retains one or more peaks reflecting the initial state.”

In the paper

“The quantum walk looks nothing like the classical random walk, it spreads out much faster in a spiky distribution that is a discrete form of an Airey function (Carteret et al. 2003).”

What the sources say

• Not checked yetcites Carteret et al. 2003

In the paper

“The stochastic matrix governing the Markov chain corresponds to the adjacency matrix in the graph-based description above.”

In the paper

“They have also found applications from the early days of quantum computation as the clock mechanism in a Feynman computer (Feynman 1986).”

What the sources say

• Not checked yetcites Feynman 1986

In the paper

“Quantum walks are simple systems in which the intricacies of decoherence can be studied both analytically and numerically, making connections with fields such as quantum chaos.”

In the paper

“Aharonov et al. (2001) proved that the coined quantum walk on a cycle with an odd number of nodes does mix to the uniform distribution, and has M( ) bounded above by O( -3 N log N), which is almost quadratically faster (in N) than a classical random walk.”

What the sources say

• Not checked yetcites Aharonov et al. (2001)

In the paper

“It is also asymmetric, with the asymmetry determined by the initial coin state.”

In the paper

“The standard deviation of P (x, T ) is √ T , that is, the walker is found on average √ T steps from the origin after T steps of the random walk.”

In the paper

“Classical random walks on a cycle mix to a uniform distribution in a time O(N 2 ).”

In the paper

“the scaling of the mixing time M( ) depends on 1/ , compared to log(1/ ) for a classical random walk”

In the paper

“The standard deviation (from the origin) σ(T ) is thus linear in T , in contrast to √ T for the classical walk.”

In the paper

“In classical random walks, the properties of the limiting distribution depend solely on the form of the graph.”

In the paper

“the simple quantum walk on a line has now been thoroughly studied”

What the sources say

• Not checked yetcites Ambainis et al. (2001)
• Not checked yetcites Bach et al. (2004)
• Not checked yetcites Yamasaki et al. (2002)

In the paper

“is known to have exact instantaneous mixing for only a few special cases, N = 3 and N = 4 (Ahmadi et al. 2003), that is, even fewer cases than are known for the discrete-time walk on the cycle as mentioned in Section 2.3.”

What the sources say

• Not checked yetcites Ahmadi et al. 2003

In the paper

“If the random walk is repeated many times, the probability distribution P (x, T ) obtained is binomial, as is well known and easily shown.”

In the paper

“Hadamard walks on cycles with an odd number of nodes converge to the uniform distribution (as was proved in Aharonov et al. (2001)), but those with an even number converge to a non-uniform distribution unless an extra phase δ is added to the Hadamard coin operator”

What the sources say

• Not checked yetcites Aharonov et al. (2001)
• Not checked yetcites Tregenna et al. (2003)
• Not checked yetcites Bednarska et al. (2003)

In the paper

“quantum random walks are deterministic, they oscillate forever, and, in general, do not mix even instantaneously.”

In the paper

“These all follow a unitary quantum evolution, apart from the final measurement.”

In the paper

“The adjacency matrix A of the N-vertex cycle graph is a circulant matrix, it has eigenvalues λ x = 2 cos(2πx/N) with corresponding eigenvectors |b x”

In the paper

“As we have seen, while there are many similarities between discrete and continuous-time quantum walks, there are also some distinctly different behaviours, and this turns out to be true also under the influence of decoherence.”

In the paper

“One of the earliest uses of non-unitary quantum walks, in Aharonov et al. (1992), was measurement-based.”

What the sources say

• Not checked yetcites Aharonov et al. (1992)

In the paper

“Spatial searching, where moving from one item in the database to the next is also counted as a cost, is also faster with a quantum algorithm, and, curiously, the continuous-time walk finds this a little harder than the discrete-time walk: in low dimensions it needs a coin too (Childs and Goldstone (2004a;2004b).”

What the sources say

• Not checked yetcites Childs and Goldstone (2004a;2004b)

In the paper

“Classical random walks also underpin many standard methods in computational physics, such as Monte Carlo simulations.”

In the paper

“Carlson et al. also identified another class of graphs that are instantaneous uniform mixing -the claw or star graphs.”

What the sources say

• Not checked yetcites Carlson et al.

In the paper

“These results can be compared with the discrete-time walk on the hypercube, which has approximate instantaneous mixing (instead of exact), and M( ) mixing in exponential time.”

In the paper

“Kendon and Tregenna (2003) took equation ( 49) and evolved it numerically for various choices of P j : projection onto the position space; projection into the coin space in the preferred basis | ± 1 ; and projection of both coin and position.”

What the sources say

• Not checked yetcites Kendon and Tregenna (2003)

In the paper

“The continuous-time quantum walk on the hypercube also hits the opposite corner in linear time, which is exponentially faster than a classical random walk”

In the paper

“Since then, several variations on quantum searching for a single item have been analysed, all essentially searching for a set of marked items of some sort (Magniez et al. 2005;Childs and Eisenberg 2005;Ambainis 2004).”

What the sources say

• Not checked yetcites Magniez et al. 2005
• Not checked yetcites Childs and Eisenberg 2005
• Not checked yetcites Ambainis 2004

In the paper

“Adamczak et al. extended their analysis of mixing in continuous-time quantum walks to a wider variety of circulant graphs, this was continued in Lo et al. (2006): mixing to non-uniform distributions turns out to be the norm for these types of graphs.”

What the sources say

• Not checked yetcites Lo et al. (2006)

In the paper

“Keating et al. (2006) applied results from localisation theory (Anderson 1958) to argue that this behaviour is exceptional, and the norm on less regular graphs is a quantum walk that tends to stay near its starting state.”

What the sources say

• Not checked yetcites Keating et al. (2006)
• Not checked yetcites Anderson 1958

In the paper

“A quantum search only needs to make O( √ N) queries, though the queries ask for many answers in superposition.”

In the paper

“The first proper algorithms using quantum walks appeared in Childs et al. (2003) and Shenvi et al. (2003), and more have followed since: for a short survey, see Ambainis (2003).”

What the sources say

• Not checked yetcites Childs et al. (2003)
• Not checked yetcites Shenvi et al. (2003)
• Not checked yetcites Ambainis (2003)

In the paper

“Randomised algorithms are one such arena, providing the best known algorithms for approximating the permanent of a matrix (Jerrum et al. 2001), finding satisfying assignments to Boolean expressions (kSAT with k > 2) (Schöning 1999), estimating the volume of a convex body (Dyer et al. 1991), and graph connectivity (Motwani and Raghavan 1995).”

What the sources say

• Not checked yetcites Jerrum et al. 2001
• Not checked yetcites Schöning 1999
• Not checked yetcites Dyer et al. 1991

In the paper

“A more systematic treatment of the quantum to classical transition in a general quantum walk appears in Kendon and Sanders (2004), which emphasises the importance of (a) demonstrating that quantum walks exhibit both wave (pure quantum) and particle (decohered) dynamics, and (b), for a non-unitary quantum walk, being able to interpolate between these two modes of behaviour by turning the decohere”

What the sources say

• Not checked yetcites Kendon and Sanders (2004)

In the paper

“Expanding the repertoire of methods for quantum algorithms was the motivation behind the recent upsurge of interest in quantum walks.”

In the paper

“Motivated by the likely form of experimental errors, they also modelled an imperfect Hadamard by applying a Gaussian spread of standard deviation √ pπ/4 about the perfect value of π/2 implicit in equation ( 8”

What the sources say

• Not checked yetcites Kendon and Tregenna (2003)

In the paper

“Starting with Shor's algorithm for factoring large numbers (Shor 1997), many of the quantum algorithms found so far belong to the same family, and are based on the use of Fourier transforms to identify a hidden subgroup (Lomont (2004) provides a recent review).”

What the sources say

• Not checked yetcites Shor 1997
• Not checked yetcites Lomont (2004)

In the paper

“This works well (that is, can be exponentially better than known classical methods) for Abelian groups, but extending the method to non-Abelian groups, where some of the notorious hard problems, such as graph isomorphism, lie, is proving tricky.”

In the paper

“They give broadly similar results for algorithmic properties such as mixing times and hitting times in those cases where both forms have been applied to the same problem, with the possible exception of spatial search (Childs and Goldstone 2004a), which, as noted above, requires a coin for maximum efficiency even in the continuoustime walk in low dimensions (Childs and Goldstone 2004b).”

What the sources say

• Not checked yetcites Childs and Goldstone 2004a
• Not checked yetcites Childs and Goldstone 2004b

In the paper

“It can be shown that a quadratic speed up is the best possible improvement for this problem (Bennett et al. 1997).”

What the sources say

• Not checked yetcites Bennett et al. 1997

In the paper

“Moore and Russell (2002) gave an analytical solution for the continuous-time quantum walks on the hypercube as well as for the discrete-time walk (see Section 2.4).”

What the sources say

• Not checked yetcites Moore and Russell (2002)

In the paper

“They show that the resulting distributions match the expected classical random walks (which may be biased if the initial state or coin operator is biased,”

What the sources say

• Not checked yetcites Košík et al. (2006)

In the paper

“The first quantum algorithm for this problem was given in Grover (1996), using a method of amplitude amplification, from which a quadratic speed up over classical searching is obtained.”

What the sources say

• Not checked yetcites Grover (1996)

In the paper

“With this choice, the continuous-time quantum walk mixes exactly instantaneously in time O(n), which is a logarithmic improvement over O(n log n) for the mixing time of a classical random walk.”

What the sources say

• Not checked yetcites Moore and Russell (2002)

In the paper

“The proof is quite technical, involving oracles and consistent colourings of the network, and requires the simulation of a continuous-time quantum walk on a discrete quantum computer.”

What the sources say

• Not checked yetcites Childs et al. (2003)

In the paper

“The converse does not apply, because the position is conditioned on the state of the coin, equation (2), thus decohering the position decohers the coin with it.”

In the paper

“However, one can do better than this: decoherence causes the quantum walk to mix to the uniform distribution without any averaging of P (x, T , p), thus retaining the logarithmic scaling with for the mixing times.It is obvious that decoherence must do this, since high decoherence rates reproduce the classical random walk, which has this property.The only question is whether it does so effectively”

What the sources say

• Not checked yetcites Kendon and Maloyer 2007
• Not checked yetcites Richter 2007b

In the paper

“In that case, mixing to the uniform distribution M( ), equation ( 15), does occur for some choices of coin operator, but this comes at the cost of requiring a number of steps linear in the inverse accuracy 1/ instead of logarithmic, unless a number of repetitions are combined in an amplification procedure.”

In the paper

“Since, as already explained, decohering the position also decoheres V. Kendon 1194 the coin, it makes only a small difference whether or not we also decohere the coin explicitly.”

In the paper

“In their study, Lopéz and Paz restrict the dynamics to cycles.However, most of their results are for small numbers of time steps such that the walk has not joined up round the cycle, and the results are identical to the walk on the line.”

What the sources say

• Not checked yetcites Lopéz and Paz (2003)

In the paper

“The first such study, Brun et al. (2003c), considered multiple coins used in sequence, with the sequence repeating after all the coins had been used once.This produces a quantum walk that still spreads linearly with the number of steps, but with the rate of spreading reduced inversely by the number of coins.”

What the sources say

• Not checked yetcites Brun et al. (2003c)

In the paper

“This brings out beautifully the fact that while decohering the coin state is sufficient to reduce a quantum walk on a cycle to a classical walk, if the walker started in a superposition of two position states, this remains untouched by the decoherence.”

What the sources say

• Not checked yetcites Lopéz and Paz (2003)

In the paper

“For all types of decoherence, the odd-N cycle shows a minimum mixing time at a decoherence rate somewhat earlier than the even-N cycle, roughly p = 2/N 2 , but because of the oscillatory nature of P (x, T , p), the exact behaviour is not a smooth function of p or .”

What the sources say

• Not checked yetcites Kendon and Tregenna 2003

In the paper

“As decoherence on the position (or both coin and position) increases, the oscillations in P (x, T , p) are damped out.At p 16/N 2 , the mixing time passes smoothly through an inflexion, and from then on behaves in a quantitatively similar manner to the adjacent-sized even-N cycles, including scaling as M min ( , p) = O(N/ ) at the inflexion.Thus for 0 < p .16/N 2 there is a region where the mixing”

What the sources say

• Not checked yetcites Kendon and Tregenna 2003

In the paper

“Decoherence on the position produces a minimum mixing time M min ( , p) = O(N/ ), thus the even-N cycle mixes to uniform in linear time for a suitable choice of decoherence rate p min ∼ 16/N 2 , independent of .”

What the sources say

• Not checked yetcites Kendon and Tregenna 2003

In the paper

“This shows that even a small amount of decoherence renders the quantum walk classical in the sense that the standard deviation of the position scales as √ t, but with a larger prefactor the smaller the dephasing: cot 2 2θ ∼ 1/θ 2 for small θ.”

In the paper

“For low rates of link-breaking, the quantum behaviour persists, while for high rates the walk makes less progress than even an unimpeded classical random walk.”

What the sources say

• Not checked yetcites Romanelli et al. (2004)

In the paper

“Their numerical results suggest the bound of H concurrent (Ω(1)) 6 O(n 2 log n 2 ) obtained by Kempe (2003b;2005) is not tight, the actual hitting time seems to be better than this in the large-n limit.”

What the sources say

• Not checked yetcites Krovi and Brun 2006a
• Not checked yetcites Kempe 2003b;2005

In the paper

“For odd-N cycles with no decoherence, they reported that M( ) ∼ N/ as compared to the upper bound of M( ) ∼ N log N/ 3 given in Aharonov et al. (2001).Richter has recently confirmed this analytically (Richter 2007b).”

What the sources say

• Not checked yetcites Aharonov et al. (2001)
• Not checked yetcites Richter 2007b

In the paper

“Brun et al. (2003b) chose pure dephasing for the form of the decoherence on the coin, so the coin projectors in equation ( 55) are”

What the sources say

• Not checked yetcites Brun et al. (2003b)

In the paper

“This calculation first appeared in Kendon and Tregenna (2002).”

What the sources say

• Not checked yetcites Kendon and Tregenna (2002)

In the paper

“The decoherence rate at the minimum mixing time is approximately π/N (odd-sized) or 2π/N (even-sized), and the mixing time itself scales as O(N log(1/ )).This thus provides a quadratic improvement over the classical mixing time, which scales as O(N 2 log(1/ )).”

What the sources say

• Not checked yetcites Kendon and Maloyer 2007
• Not checked yetcites Richter 2007b

In the paper

“Although for N divisible by 4, the coin-decohered mixing time shows a minimum below the classical value at p 2/N, this mixing time is & N 2 /32 , that is, it is still quadratic in N. Thus, although noise on the coin causes the even-N cycle to mix to the uniform distribution, it does not produce a significant speed up over the classical random walk.”

What the sources say

• Not checked yetcites Kendon and Tregenna 2003

In the paper

“The overall conclusion is thus the same as for the walk on a line: there is a useful window within which decoherence enhances rather than degrades the quantum features of the walk.Moreover, it is actually necessary to ensure that the scaling with precision is efficient.Additionally, decoherence ensures the quantum walk mixes to a uniform distribution, regardless of the initial conditions or choice”

In the paper

“Richter (2007b) actually proves not only that the optimal mixing time is O(N log(1/ )), but also that three different strategies for optimising the mixing time are equally effective.”

What the sources say

• Not checked yetcites Richter 2007b

In the paper

“These effects are not algorithmically significant, that is, they do not alter the scaling of quantum processing, but, nevertheless, may be useful in practice, and represent intriguing and complex behaviours worthy of study in their own right.”

In the paper

“In contrast, determining whether a directed graph allows a unitary dynamics that respects the graph structure is a hard problem, to which a general answer is not known.”

In the paper

“In other words, the walk suffers a quantum Zeno effect (Misra and Sudarshan 1977), a phenomenon that does not appear in coined quantum walks.One consequence of this is that quantities such as the concurrent hitting time are infinite, since the continuous monitoring will ensure the continuous-time quantum walk never arrives.”

What the sources say

• Not checked yetcites Misra and Sudarshan 1977

In the paper

“When no decoherence is present, the walk hits at H one-shot (1) = nπ(2c + 1) 2k , and it does so exactly, that is, r(c) = 1.This corresponds to the result for the discrete-time quantum walk given in Kempe (2003b;2005), see Section 5.6, so both discrete-time and continuous-time quantum walks on the hypercube exhibit exponentially faster hitting times than classical random walks.The difference is”

What the sources say

• Not checked yetcites Kempe (2003b;2005)

In the paper

“We now show that with the model of decoherence described above each dimension still behaves independently.Recall equation ( 41) showing that the adjacency matrix for the hypercube decomposes into a sum of tensor products, with each acting only on a single qubit.Since the noise operators P j also have this structure, provided ρ(0) starts off in a state that is also decomposable (such as the all”

What the sources say

• Not checked yetcites Alagić and Russell (2005)

In the paper

“Dür et al. (2002) proposed a quantum walk on the line using an atom hopping between traps in an optical lattice.”

What the sources say

• Not checked yetcites Dür et al. (2002)

In the paper

“Decoherence in discrete-time walks has thus provided us with a number of notable common features, such as smoother spreading, enhanced mixing, and a transition from under-damped (quantum) to over-damped (classical) behaviour, which is analogous to classical damping in a harmonic oscillator.”

In the paper

“We have observed some of the same effects of enhanced mixing and smoother spreading, and also shown that where the decoherence does not directly help, the quantum speed up is at least robust under small amounts of decoherence.”

In the paper

“This is actually a quantum computation of a quantum walk, since the three qubits are used to represent the binary number labelling the vertex (two qubits), and the qubit coin.”

What the sources say

• Not checked yetcites Ryan et al. 2005

In the paper

“Sanders et al. (2003) suggested a quantum walk using the phase of the electromagnetic field in a cavity as the position (so the walk is again on a cycle) with the role of the coin played by an atom”

What the sources say

• Not checked yetcites Sanders et al. (2003)

In the paper

“The concurrent hitting time already includes a portion of decoherence (no extra features are produced by the addition of unselective decoherence), but there is still a range of 0 < p .1/n within which the quantum speed up is preserved.Note that in both the one-shot and concurrent cases, p 1/n is a critical damping rate, smoothing out the second peak (shown at around 40 (≡ 3nπ/2) steps in Figure 10”

What the sources say

• Not checked yetcites Kendon and Tregenna 2003

In the paper

“Starting with an undirected graph, a corresponding directed graph can be constructed by a mapping due to Watrous (2001), which is similar to the line graph definition above, except that each undirected edge is treated as a pair of directed edges.”

What the sources say

• Not checked yetcites Watrous (2001)

In the paper

“As p → ∞, the walk suffers from the quantum Zeno effect.Informally, we can say that the rate of decoherence is so large that the walk is continuously being reset to the initial wave function |0 ⊗n by measurement.By inspection of equations (119), it is clear that P [0] → 1 and P [1] → 0 as p → ∞.”

What the sources say

• Not checked yetcites Alagić and Russell (2005)

In the paper

“On the face of it, M( ) appears to scale as log(N), which would imply an exponential speed up over classical mixing times of O(N 2 ).However, this formula is only valid for pN 1, so the 1/p scaling ensures that M( ) N for the range of validity of the result.What this result tells us is that decoherence causes the continuous-time quantum walk to mix, with the dominant effect a scaling of 1/p, and”

In the paper

“By examining the local maxima of P [1], we can determine that the walk has approximate instantaneous hitting times to the opposite corner (1, . . ., 1) at times H one-shot (r) = 2πn(2c + 1) 16k 2 -p 2(121) for all c ∈ Z, c > 0. However, the probability r(p, c) of measuring an exact hit decays exponentially: r(p, c) = 1 2 + 1 2 exp (-pπ(2c + 1)/ 16k 2 -p 2 ) n . (122 )”

What the sources say

• Not checked yetcites Alagić and Russell (2005)

In the paper

“These bounds show that M( ) is linearly proportional to the decoherence rate p, but is quadratically dependent on N, and logarithmically dependent on .The change in the form of the scaling with p from reciprocal for small p to linear for large p suggests there is a minimum mixing time at some intermediate decoherence rate.Fedichkin et al. (2006) confirmed this numerically: see Figure 12.”

What the sources say

• Not checked yetcites Fedichkin et al. (2006)

In the paper

“In fact, the proofs in Richter (2007b) are sufficiently general that they apply equally to the continuous-time walk as to the discrete-time walk.This fills the gap between small and large decoherence rates, and proves that continuous-time walks, with O(log(1/ )) decoherence or measurement events, mix in time O(N log(1/ )) on the cycle (and the d-dimensional torus).”

What the sources say

• Not checked yetcites Richter 2007b

In the paper

“Fedichkin et al. (2006) gives the scaling for mixing times for the walk on the cycle for both small and large decoherence rates.They augment this with numerical studies showing a minimum mixing time for intermediate decoherence rates.”

What the sources say

• Not checked yetcites Fedichkin et al. (2006)

In the paper

“physical systems can be made to perform a quantum walk, and this process is of interest in its own right as an example of precise coherent control.”

In the paper

“Travaglione and Milburn (2002) described how a coined quantum walk can be performed using the vibration modes of the trapped ion coupled to an internal state of the ion as the quantum coin.”

What the sources say

• Not checked yetcites Travaglione and Milburn (2002)