Time evolution analysis - Entangling gate

7.3 Entangling gate

7.3.2 Time evolution analysis

L (7.29)

M (k_i) = 2t₀sin|ki| = µi. (7.30)

With these choices, a wave-packet moving with speed sin|k1| travels a distance Z + 4L ≈ (11 + α)L in approximately the same time that a wave-packet moving with speed k₂takes to travel a distance Z + 2D + 2L≈ (9 + α)L + 2D, since

t_Cθ = (11 + α)L

2 sin|k1| ≈ (9 + α)L + 2D

2 sin|k2| . (7.31)

Additionally, these distances are chosen so that at time t₁ = (4 + α)L/2 sin|k1|, the two particles are both located on the center path, with the wave-packet with momentum k₁ a distance (1 + α)L from the first momentum switch, the other a distance L from the second momentum switch, and the two wave-packets a distance L apart, and so that at a time t₂= t₁+ 6L/(sin|k1| + sin |k2|), the wave-packets have passed each other, but are now the same distance from the other momentum switch. This is represented pictorially in Figure 7.4.

One point to notice is that the distances M (k_i) = µ_i are chosen to be the distance traveled by each particle over some time t₀. As such, if at time−t0 the two particles were both centered along the endpoints of the paths, they would . This isn’t particularly important, except to allow us to nicely combine this result with other scattering events.

7.3.2 Time evolution analysis

With the graph G well defined for two momenta k₁ and k₂ and cut-off distance L, we now want to actually analyze the dynamics of the system. We want to claim that the initial states evolve into encoded logical output states, where the acquired phases correspond to an entangling gate.

In particular, we will want that the evolution of an encoded state on this graph corresponds to some unitary diagonal in the computational basis, in which an additional phase is acquired for the state |11i. While in an ideal case, this will simply be a controlled-θ gate, in which the applied unitary is the identity for the other three basis states, our graph will also have additional phases

(a) k₁ →

k₂ → M (k₁) L L

M (k₂) L L + D

(b)

k₁ ↓ k2↑

(1 + α)L 2L

L 2L

(c)

k₁ ↓

k2 ↑ 1L

2L L

2L(1 + α)L

(d) k2 →

k₁ → L + D L M (k2)

L L M (k₁) Figure 7.4: This picture illustrates the scattering process for two wave-packets that are incident on the input paths as shown in figure (a) at time t = 0. Figure (b) shows the location of the two wave-packets after a time t₁ and figure (c) shows the wave-packets after a time t₂. After the particles pass one another they acquire an overall phase of e^iθ^±. Figure (d) shows the final configuration of the wave-packets after a total evolution time t_Cθ.

corresponding to a product of single-qubit unitaries, arising from the transmission coefficients of the two momentum switches.

However, in order to apply an encoded unitary, we will need to have an encoding of the logical output states. Letting t_Cθ be as defined in(7.31), we will have that the encoded output states for each qubit will be

|0outi¹= γe^−2it^Cθ^cos(k¹⁾

νX1+L x=ν1−L

e^ik¹^xe⁻^(x−ν1)

2σ2 |x, 1i (7.32)

|1outi¹= γe^−2it^Cθ^cos(k¹⁾

νX1+L x=ν1−L

e^ik¹^xe⁻^(x−ν1)

2σ2 |x, 2i (7.33)

|0outi²= γe^−2it^Cθ^cos(k²⁾

νX2+L x=ν2−L

e^ik²^xe⁻^(x−ν2)

2σ2 |x, 4i (7.34)

|1outi²= γe^−2it^Cθ^cos(k²⁾

νX2+L x=ν2−L

e^ik²^xe⁻^(x−ν2)

2σ2 |x, 3i (7.35)

where ν1 = µ1 +d2tCθsin|k1|e and ν2 = µ2 +d2tCθsin|k2|e. Intuitively, these are simply the expected positions of the wave-packets corresponding to the initial logical states after a time t_Cθ, along with the phases acquired from their energies.

Additionally, note that the eigenstates of H_G² decompose into a symmetric and an antisymmetric subspace. While it is intuitively more clean to work with distinguishable particles, the phases

acquired during the evolution on G will depend on the symmetry of the underlying particles. As such, let us define the eight states

|(z1z₂)_in/outi±= 1

√2

|(z1)_in/outi¹|(z2)_in/outi²± |(z2)_in/outi²|(z1)_in/outi¹

. (7.36)

Our analysis will be for these states.

If we note that the input states don’t particularly rely on the initial distance from the graph, and that the output states similarly don’t rely on the distance from the end, we will be able to easily combine the time evolution on these graphs with the time evolution on other graphs. Just so long as the initial wave-packets start a distance proportional to L away from the ends, and similarly end a distance proportional to L from the ends, we can usLemma 1in order to determine the overall time evolution.

In particular, we have the following lemma:

Lemma 17. Let G⁰ be the graph given in Figure 7.3 with distances given by (7.25)–(7.27), and where we assume that the initial and final states as defined in (7.23), (7.24), and (7.32)-(7.35) only have support on vertices a distance at least L/3 from the ends of the truncated paths. If the momentum switch has transmission coefficientT₁ for momentumk₁ and transmission coefficientT₂ for momentumk₂, and if the phase acquired for two particle scattering between momentumk₁andk₂ is given by θ_± for symmetric and antisymmetric states, then we have the following approximations for the time-evolved states:

Proof. The first three bounds (7.37), (7.38), and (7.39) are similar to the proofs of Chapter 5, since in each case the two particles are supported on disconnected subgraphs and thus we can use Lemma 13. For each of the single-particle scattering events, we can use a strategy similar to Lemma 12to bound the error in our time-evolution for the unsymmetrized two-particle states. To get the bound of(7.40), the proof strategy will be similar, but we will also need an application of Theorem 6 during the times at which both particles are located on the long path.

Let us first understand the single-particle evolutions on the long paths. Note that t_Cθ ≤ cL for some constant c, and thus according to Theorem 4, on an infinite path P we have that

for some constant ξ and where µ(t) = µ+d2t sin(k)e. WhileTheorem 4doesn’t exactly give us this, if we apply the theorem to a graph gadget bG that results in the final graph being a long path, and assume that the initial location is far from the scattering event, the result follows after a relabeling

of the basis states. If we then note that the corresponding approximation involving µ(t) on the finite path always remains a distance at least L/3 away from the ends of the finite approximation, Lemma 1with H = A(P ), ˜H = A(G⁰), and N₀ = ^L₃, and the error bound of δ = ξp

log L/L gives us

e^−iA(G⁰^)t^Cθ|0iniⁱ− |0outiⁱ ≤

8et_Cθ N₀ + 2

rlog L

L + 2^−N⁰

1− ξ

rlog L L

(7.42)

≤ ζ

rlog L

L (7.43)

for some constant χ that is independent of the momentum k. Hence, we have that this approxima-tion holds for both long paths.

Now let us analyze the single-particle evolution on the subgraph that connects the two particles.

For both particles, we will define approximations to the time-evolved wave-packets that will equal both the initial state and the corresponding final states. Namely, if we also label the vertices of the path 5 (the one of length Z which are currently labeled (x, 5)) as (W +1+x, 2) and (X +Z−x+2, 3), then we can define the states

|α¹1(t)i = γe^{−2it cos(k}¹⁾

µ(t)+LX

x=µ(t)−L

e^ik¹^xe⁻^(x−µ(t))2^2σ2 |x, 2i (7.44)

|α¹2(t)i = γe^{−2it cos(k}²⁾

ν(t)+LX

x=ν(t)−L

e^ik²^xe⁻^(x−ν(t))2^2σ2 |x, 3i (7.45)

where µ(t) = µ +d2t sin(k1)e and ν(t) = ν + d2t sin(k2)e. Note that these are (almost) the same approximations as used for the long paths, where we assume that the momentum switches essentially act as connections between the correct long paths.

With these approximations, we will then want to show that the time evolution of the initial states approximately follow these output states, possibly with some additional phases. In particular, note that there are three times (not including t = 0) that are important for our evolution,

t₁ = (4 + α)L

2 sin|k1| (7.46)

t2 = t1+ 6L

sin|k1| + sin |k2| (7.47)

t_Cθ = (11 + α)L

2 sin|k1| . (7.48)

If we can show that the time-evolved approximation at one time is close to the approximation at the next time for each of these times, we then have that our final approximation and the time-evolved initial state are close as well. To do so, we will use the same procedure for combining our time-evolution on graphs as was done in Chapter 5.

Namely, we can use Lemma 1 and Theorem 4 to analyze the evolution of these states. Note that Theorem 4 gives us an approximation for the evolution of |α¹_j(t)i via the momentum switch on the infinite path, and two applications of Lemma 1 allows us to then analyze the evolution of the state for times 0 ≤ t ≤ t1 (after a suitable relabeling of the vertices). In particular, we have that

e^−iH⁽¹⁾^G0^t¹|α¹j(0)i − Tj|α¹j(t₁)i ≤ ζj¹

rlog L

L , (7.49)

where the dependence of the constant on j is only there in terms of the initial distance of the wave-packet from the ends of the finite paths.

We can then use the same trick, namely two applications ofLemma 1and a single application of Theorem 4 to understand the evolution on the path of length Z between times t₁ and t₂. Literally the same analysis as in the previous paragraph again gives us that

e^−iH^G0⁽¹⁾^(t²^−t¹⁾|α¹j(t₁)i − |α¹j(t₂)i ≤ ζj²

rlog L

L (7.50)

where again the constant only depends on the distance between the support of the approximation and the truncated ends, but in both cases is a constant.

Finally, a third application of this trick around the second momentum switch gives us

e^−iH^G0⁽¹⁾^(t^Cθ^−t²⁾|α¹j(t₂)i − Tj|α¹j(t_Cθ)i ≤ ζj³

rlog L

L , (7.51)

where we again have used a necessary relabeling of the vertices.

Combining the three bounds (7.49)–(7.51), we then have that

e^−iH^G0⁽¹⁾^(t^Cθ⁾|α¹j(0)i − Tj²|α¹j(tCθ)i ≤ (ζj¹+ ζ_j²+ ζ_j³)

rlog L

L , (7.52)

and thus we have approximations to the single-particle evolutions.

From this, if we use Lemma 13 along with these three bounds on the evolution for times t_Cθ, we have bounds similar (7.37), (7.38), and (7.39) for the unsymmetrized states. However, since the bounds (and approximations) don’t depend on the symmetrization of the underlying states, we also have the symmetrized and antisymmetrized bounds as well.

Finally, we need to show the bound(7.40). Unfortunately, this bound is slightly more difficult to prove, as we cannot naively useLemma 13. However, a more nuanced application ofLemma 1will allow us to approximate the evolution of both particles on G⁰ by the evolution on a disconnected graph for times less that t₁and larger than t₂. We will then only need to work with the two-particle interactions for times between t₁ and t₂, and here we will be able to useLemma 1to approximate the analysis by that on an infinite path for both the symmetric and anti-symmetric states. Putting everything together, we have a good approximations for all times 0 < t < t_Cθ, and thus we also have a good approximation for the final state.

To do this, let us define the graph G^sep₁ to be the G⁰ with the vertex labeled (d(3+α+1/2)Le, 5) removed and let G^sep₂ be G⁰ with the vertex labeled (d(3 + 1/2)Le, 5) removed. Note that both G^sep_i have four components, whereas G⁰ had three, but importantly that the two particles only have amplitude on separate components. As such, we can then use Lemma 13 and Lemma 1 as in the case for any of the other logical computational basis states, at least for times in which the single-particle wave-packets remain far from the removed graph. In particular, for all times 0 < t < t₁, the approximation arising from the corresponding two-particle evolution on disconnected graphs (i.e., both the single-particle evolutions) remain a distance at least L/3 from the removed vertex.

Hence, usingLemma 13and the same single-particle evolutions as above, we have that

e^−iH^G0⁽²⁾^t¹|11ini − T1T2|α¹1(t1)i|α²1(t1)i ≤ ξ³

rlog L

L (7.53)

for some constant ξ₃. In particular, we have that each of the individual single-particle evolutions an the finite-sized graphs have errors bounded by ξ₁ and ξ₂timesp

log L/L (as in the above examples),

and thus the two-particle evolution on G^sep has an error bound with constant ξ₁+ ξ₂. Another application ofLemma 1, using the fact that we are dealing with two-particles and thus the norm of H_G⁽²⁾0 is bounded, the approximations are far from the removed vertex, and our bound on the error when evolving on G^sep, then gives the error as claimed.

At this point, our approximations to the time-evolved wave function are only supported on the long path far from the ends of the path, and thus we can use two applications ofLemma 1to look at the time-evolution on an infinite path. This is exactly the reason for Theorem 6, and we can thus use it’s bounds. It is at this point that we discover different evolutions for symmetric and anti-symmetric particles, as the relevant phase acquired during the overlap is dependent on the symmetry between the particles.

In particular, if P is the infinite path with vertices corresponding to the finite path of length Z, we have that Theorem 6gives us that

e^−iH

P2(t2−t1)

|φ^L,σ_d(2+α)Le(−k1)i|φ^L,σ_d(7+α)Le(k₂)i ± |φ^L,σ_d(7+α)Le(k₂)i|φ^L,σ_d(2+α)Le(−k1)i

− e^iθ^±e^−2i(t²^−t²^{)(cos k}¹^{+cos k}²⁾

|φ^L,σ_4L(−k1)i|φ^L,σ_2L(k₂)i ± |φ^L,σ_2L (k₂)i|φ^L,σ_4L (−k1)i

≤ ξ4

L, (7.54)

with an approximation for all intermediate times that has a similar bound. With two applications of Lemma 1, along with a relabelling of vertices, we can see

e^−iH⁰⁽²⁾^G ^(t²^−t¹⁾|α¹1(t₁), α¹₂(t₁)i±− e^iθ^±|α¹1(t₂), α¹₂(t₂)i±

≤ ζ4

rlog L

L . (7.55)

After the two particles have moved passed each other and we have reach time t₂, we can use the same trick to approximate the time evolution for t2 < t < t_Cθ as for the early times, except on the graph G^sep₂ . We find that the error in our approximation for these times is bounded by the same value as for the early times, as the analysis is nearly identical.

If we then put together these three bounds, we have:

e^−iH^G⁰⁽²⁾^t^Cθ|11ini^1,2_± − e^iθ^±T₁²T₂²|11outi^1,2_± (7.56)

≤ e^−iH^G⁰⁽²⁾^t¹|α¹1(0), α¹₂(0)i±− T1T2|α¹1(t1), α¹₂(t1)i±

+ e^−iH⁰⁽²⁾^G ^(t²^−t¹⁾|α¹1(t₁), α¹₂(t₁)i±− e^iθ^±|α¹1(t₂), α¹₂(t₂)i±

+ e^−iH⁰⁽²⁾^G ^(t^Cθ^−t²⁾|α1¹(t₂), α¹₂(t₂)i±− T1T₂|α¹1(t_Cθ), α¹₂(t_Cθ)i±

(7.57)

≤ ξ5

rlog L

L (7.58)

and we have (7.40).

If we then take χ to be the maximum constant for our four bounds, we have proven the theorem.

At this point, we now understand how our assumed initial state propagates for single- and two-particle states, and we have the necessary building blocks for our universality result.

Note that if θ₊(k₁, k₂) = θ₋(k₁, k₂), then our results from above hold for distinguishable particles as well, since the applied unitary is then the same for all linear combinations of the two states.

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