Methods for classically simulating noisy networked quantum architectures

An IQP machine is defined by its capacity to implement X-programs and sample from the output distribution.

Definition 3.1 An X-program consists of a Hamiltonian comprised of a sum of products of X operators on different qubits, and θ ∈ [0, 2π] describing the time for which it is applied. The $h\mathrm{th}$ term of the sum has a corresponding vector ${{\bf{q}}}_{h}\in {\left\{0,1\right\}}^{{n}_{a}}$, called a program element, which defines on which of the n_a input qubits, the product of X operators which constitute that term, acts. The vector ${{\bf{q}}}_{h}$ has 1 in the $j\mathrm{th}$ position when X is applied on the $j\mathrm{th}$ qubit. As such, we can describe the X-program using θ and the matrix ${\bf{Q}}=({{\bf{Q}}}_{{hj}})\in \{0,1\}{}^{{n}_{g}\times {n}_{a}}$ which has as rows the program elements ${{\bf{q}}}_{h}$, $h=1,\,\ldots ,\,{n}_{g}$.

Applying the X-program defined above to the state $\left|{0}^{{n}_{a}}\right\rangle$ and measuring the result in the Z basis produces the following probability distribution, ${\mathtt{X}}$, of outcomes:

$$\begin{eqnarray}&&{\mathbb{P}}\left({\mathtt{X}}=\widetilde{x}\right)={\left|\left\langle \widetilde{x}\right|\exp \left(\displaystyle \sum _{h=1}^{{n}_{g}}{\rm{i}}\theta \mathop{\displaystyle \bigotimes }\limits_{j:{{\bf{Q}}}_{{hj}}=1}{X}_{j}\right)\left|{0}^{{n}_{a}}\right\rangle \right|}^{2},\ \ \widetilde{x}\in \{0,1\}{}^{{n}_{a}}.\end{eqnarray} \tag{ 1 }$$

Definition 3.2 Given an X-program, an IQP machine is any computational method capable of efficiently returning a sample $\widetilde{x}$ from the probability distribution of equation (1).

IQP involves only gates which are diagonal in the Pauli-X basis so does not achieve the full power of quantum computation. However, it is believed to be hard to classically simulate. Below we consider weak simulation of a circuit family which is when, given a circuit's description, its output distribution can be sampled from by a polynomial time classical computer.

Theorem 3.1 If the output probability distributions generated by uniform families of IQP circuits could be weakly classically simulated then the polynomial hierarchy (PH) [58] would collapse to its third level.

A collapse of PH is thought to be unlikely, giving us confidence in the hardness of IQP.

While theorem 3.1 is a worst case hardness result, we can trim some instances form the set of problems we would expect to demonstrate quantum-advantage. For example [42] when $\theta \in \left\{\tfrac{\pi n}{4}:n\in {\mathbb{Z}}\right\}$ the result of the computation is classically computable. In the protocols we set $\theta =\tfrac{\pi }{8}$, giving us the necessary hardness.

Theorem 3.1 and similar results in [54] are remarkable in their demonstration that quantum computers which are very much weaker than a universal BQP machine are impossible to classically simulate. These results are, however, proven in the setting where one demands a classical simulator produce samples which are within a multiplicative error, which depends on the probability of the sample, of the ideal quantum distribution. It is more realistic, and closer to the true capabilities of noisy quantum computers, to allow the classical simulator to be wrong up to an additive error. That is to say that the device need not necessarily sample from the ideal distribution P, but any distribution $\widetilde{P}$ with the property

$$\begin{eqnarray*}&&\displaystyle \sum _{x\in {\left\{\mathrm{0,1}\right\}}^{{n}_{a}}}\left|\widetilde{P}\left(x\right)-P\left(x\right)\right|\leqslant \epsilon \end{eqnarray*}$$

for some constant . This measure of distance between distributions is also called the ℓ₁-norm distance. In this case too, hardness results exists.

Theorem 3.2 Assume either one of two conjectures, relating to the hardness of Ising partition function and the gap of degree 3 polynomials, and the stability of the PH, it is impossible to classically sample from the output probability distribution of any IQP circuit in polynomial time, up to an additive error of $\epsilon =\tfrac{1}{192}$.

We will take the hardness of weak simulation up to additive error as an indication that a class of problems is promising for an early demonstration of quantum-advantage. This is justified because it seems plausible that noise will have a similar impact on average case problems, which we simulate, and worst case problems, for which hardness results exist. Thus we can draw conclusions about the impact of noise on the hard cases from its impact on average cases.

Ideally, we would like for our class to demonstrate an advantage in the average case as proofs of these results are often constructive, and would present us with schemes to implement. Such results, of which the following is an example, are harder to obtain, especially if one requires noise tolerance and architectural restrictions.

Theorem 3.3 Assuming the integrity of PH and the difficulty of approximating an Ising model partition function; there is a family of IQP circuits, implemented in depth $O(\sqrt{n}\mathrm{log}n)$ on a 2D square lattice and containing $O(n\mathrm{log}n)$ 2-qubit gates, for which a constant fraction of circuits cannot be simulated classically.

Here the simulation is understood as a simulation up to an additive error. This 2D square lattice architecture is favoured by many quantum computers today [59, 60] and while we hope to be impartial to the architecture [61–63], for early devices it is important to engineer our tests with this in mind. However, it is likely that the qubit routing used to implement the circuit of theorem 3.3 on a square lattice requires many swap gates. These would not commute with the rest of the circuit, destroying the instantaneous nature which, as we will see, we prefer for our purposes.

Theoretical studies of quantum-advantage in the presence of noise have also explored the following, arguably more realistic, settings. The first considers independent depolarising noise which is added to all qubits at the end of the circuit. In this case the noise per qubit does not, as in the additive case, depend on the number of qubits. It is shown [35] that the circuit family of theorem 3.3 are classically simulable in this noise model but that classical hardness can be recovered by modifying the circuits to include some classical error correction technique. Second, the more general case of independent noise being applied to each gate also leads to a wide family of circuits becoming classically simulable [64].

In our work, we do not explore the impact of noise on the quantum-advantage at a theoretical level, as was done in the aforementioned works, but suggest that numerical exploration should be done in parallel with the theoretical analysis. This would guide us in understanding which realistic experimental setting is best to demonstrate quantum-advantage with IQP problems.

A common framework for studying quantum computation is the Measurement-Based Quantum Computation (MBQC) model [65–67]. Problems in the IQP class admits a realisation, using MBQC, which is particularly useful since it explicitly parallelises the computation.

The MBQC implementation of a given X-program uses a graph state defined by a corresponding bipartite graph.

Definition 3.3 We define the bipartite graph of an X-program $({\bf{Q}},\theta )$, as the graph with biadjacency matrix ${\bf{Q}}=({{\bf{Q}}}_{{hj}})\in \{0,1\}{}^{{n}_{g}\times {n}_{a}}$. This means that there is a bipartition of vertices into two sets A and G of cardinality n_a and n_g and that an edge exists in the graph between vertex g_h of set G and vertex a_j of set A when ${{\bf{Q}}}_{{hj}}=1$. The sets of vertices $G=\left\{{g}_{1},...,{g}_{{n}_{g}}\right\}$ and $A=\left\{{a}_{1},...,{a}_{{n}_{a}}\right\}$ will be called gate and application vertices respectively. See figure 2 for an example.

Zoom In Zoom Out Reset image size

One can prove [42] that the distribution of equation (1) can be achieved by initialising n_a application qubits in the states $\left|{a}_{j}\right\rangle =\left|\,+,\,\right\rangle$, n_g gate qubits in the states $\left|{g}_{h}\right\rangle =\left|\,+,\,\right\rangle$, applying Controlled-Z operations between qubits when there is an edge in the bipartite graph described by the X-program matrix ${\bf{Q}}$ and measuring the resulting state. The measurement of the application qubits is in the Hadamard basis, and of the gate qubits is in the basis of equation (2).

$$\begin{eqnarray}&&\left\{\left|{0}_{\theta }\right\rangle ,\left|{1}_{\theta }\right\rangle \right\}=\left\{\displaystyle \frac{1}{\sqrt{2}}\left({{\rm{e}}}^{-{\rm{i}}\theta }\left|+\right\rangle +{{\rm{e}}}^{{\rm{i}}\theta }\left|\,,-,\,\right\rangle \right),\displaystyle \frac{1}{\sqrt{2}}\left({{\rm{e}}}^{-{\rm{i}}\theta }\left|+\right\rangle -{{\rm{e}}}^{{\rm{i}}\theta }\left|\,,-,\,\right\rangle \right)\right\}.\end{eqnarray} \tag{ 2 }$$

The measurement bases do not depend on the outcomes of other measurements and therefore can be parallelised to one round of entanglement and measurement.

Importantly the distribution of equation (1) is achieved via this implementation in polynomial time. As such the complexity results of section 3.1.2 apply here.

In [68] a subclass of IQP problems called 2-dimensional dynamical quantum simulators (2D-DQS) are defined. The name references the 2D square lattice architecture involved and that they could be realised with sub-universal quantum simulators. Architecture I from [68] is seen in Protocol 1.

Protocol 1. A description of an instance of the 2D-DQS problem introduced by [68]. E and V are the edge and vertex set respectively of a ${N}_{x}\times {N}_{y}$ 2D square lattice.

1: Choose $\tau \in {\left\{0,1\right\}}^{{N}_{x}\times {N}_{y}}$ uniformly at random.

2: Initialise the product state: $$\begin{eqnarray}&&\left|{\phi }_{\tau }\right\rangle =\underset{i=1}{\overset{N={N}_{x}\times {N}_{y}}{\displaystyle \bigotimes }}\left(\left|0\right\rangle +{e}^{i{\tau }_{i}\tfrac{\pi }{4}}\left|1\right\rangle \right)\end{eqnarray} \tag{ 3 }$$ 3: Allow system to evolve for time $t=1$ according to the nearest neighbour, translation invariant, Ising Hamiltonian: $$\begin{eqnarray}&&H:= \displaystyle \sum _{\left(i,j\right)\in E}\displaystyle \frac{\pi }{4}{Z}_{i}{Z}_{j}-\displaystyle \sum _{i\in V}\displaystyle \frac{\pi }{4}{Z}_{i}\end{eqnarray} \tag{ 4 }$$   This is equivalent to applying controlled Z operations on each edge.

4: Measure all qubits in the X basis.

The construction is summarised in figure 3. One realises that this is within IQP by noting either that it is simply a Bloch sphere rotation of the definition in section 3.1.1 or that it is a constant depth commuting circuit on a 2D lattice. The related hardness result for this architecture is seen in theorem 3.4.

Zoom In Zoom Out Reset image size

Theorem 3.4 Assuming three conjectures (one being the non-collapse of PH), a classical computer cannot sample from the outcome distribution of the architecture of Protocol 1 up to an additive error of $\tfrac{1}{22}$ in time polynomial in ${N}_{x},{N}_{y}$.

We note that this problem seems a good candidate for our purposes, as described in the hard problem selection methodology of Step 1 in section 2.1, since it is hard to simulate classically and is experimentally realisable in the near term. A further advantage of this scheme is that the authors of [68] provide an explicit means for a client with a simple measurement device to verify the protocol. This is an important feature for extending the analysis beyond the limits were classical simulation is possible.

• Depolarising. Caused by scattering of amplitudes of the electron's wave-function between different energy levels of the ion. Modelled by a random single-qubit Pauli on each qubit at a rate of $\approx 9\times {10}^{-4}\,{{\rm{s}}}^{-1}$.
• Dephasing. Entanglement reduction that destroys data not stored in the standard basis. Modelled by Z gate on each qubit at a rate of $\approx (7.2\pm 1.4)\times {10}^{-3}\,{{\rm{s}}}^{-1}$.

To simulate these noise channels we need the execution times of different operations:

• Preparation—1–1.5 ms.
• Measurement—2–2.5 ms.
• Single or two-qubit operation within a trap −0.5 ms.¹⁴
• Linking between traps −1–2 s.¹⁵

• Preparation. Error probability in preparing a state. Modelled by Pauli X at rate of $\approx 2\times {10}^{-4}$.
• Measurement. Similarly to preparation, measurement is also noisy. Rate of $\approx 5\times {10}^{-4}$ to measure incorrectly any qubit, which corresponds to an X gate.
• Single-qubit. Gates random Pauli operator applied in addition to the single-qubit gate with probability $\approx (1.5\pm 0.45)\times {10}^{-6}$.
• Two-qubit gates. Modelled by independent single-qubit random Pauli errors on both qubits, each with probability $\approx (5.5\pm 3.5)\times {10}^{-4}$ and a further two-qubit error $Z\otimes Z$ with probability $\approx 6\times {10}^{-5}$.
• Linking operations. Depending on the amount of entanglement distillation used [72], this error varies since it is determined by the fidelity of the entanglement. If 10-qubits are used for distillation, then the effect is approximately the same as the regular (same ion-trap) two-qubit gate [73]. Moreover, using more qubits for distillation would not improve the computation since the same ion-trap qubit gates will still have higher errors.

This noise description is specific to the NQIT Q20:20 device. However, the structure is general and other versions of the NQIT device or other quantum devices are likely to have similar 'specifications'. Therefore the toolbox developed should be adaptable to other quantum computation devices. The reader may refer to appendix A.2 for a systematic description of the noise.

We consider Architecture I from [68] as discussed in section 3.1.4 and constrain it according to the noise of the NQIT machine as listed in section 3.3. For simplicity, we will use a modified version of the NQIT architectural restraints of section 3.2.

Constraints. The 2D-DQS problem has been designed for networked architectures and, with some simple adaptations, it can coincide with NQIT's device. In particular, by making the simplifying assumption that we use a single logical qubit per ion-trap¹⁸ we can map every grid vertex onto a single ion trap. One may then look to figures 3 and 4 to understand that the 2D-DQS problem can be easily overlaid onto the NQIT architecture, which also permits the necessary measurements, state preparations, and single and 2-qubit gates.

As the adapted NQIT architectural restrains, detailed above, adhere to those required for the 2D-DQS problem seen in Protocol 1, the worst case additive error hardness result of the 2D-DQS problem, as seen in theorem 3.4, applies. While we have agreed that this setting constitutes one that is worthy of investigation, as the noise levels are independent for each qubit and not dependent on the problem size, the additive error permitted by theorem 3.4 is likely exceeded. Hence, we would expect that in the noisy case the distribution becomes far from the perfect one and for the advantage to diminish.

Simulation. We consider $4\times 5$ grids, modelling 20 ion traps in total, and use them to perform Protocol 1. Protocol 1 requires, on average, half as many T gates as qubits; in this case 10 and 20 respectively. Details of the numerical specifics of the experiments can be found in appendix B.2. Here it suffices to say that we use four steps to generate the entangled 2D cluster. The number of steps plays a role in the amount of noise as it determines the duration of the computation and thus the decoherence time we consider.

We perform 20 trials, each concerning one perfect circuit and a random output string. For each trial there are 20 noisy runs, each with their own random noisy version of the trial's circuit. This random noisy version of the perfect circuit is generated by considering the noise type and strength of the experiment as described in appendix B.2. We simulate all 21 circuits 20 times, calculating the mean probability of measuring the corresponding bit string in each case. We will then take the mean and standard deviation of the noisy runs.

While, as noted in [74], simulations of up to 40 qubits and 50 T gates is possible using this simulator, as is also noted in that work, doing so takes several hours. In our case we simulate 20 trials, each with 21 runs and 20 simulations per run and so we restrict the number of qubits and T gates to a more manageable amount. Later in this work we go further and perform many thousands of simulations in each numerical experiment, justifying our restriction.

Figure of merit. For this numerical experiment we will utilise the 'accuracy and far from uniformity of noisy runs' condition from the introduction to section 4. In particular, we will consider a perfect run to be far from uniform when it is either greater than twice the uniform value, or less than half. In this way we will identify if the noise level reveals that, as we expect, the potential for a demonstration of quantum-advantage should be dismissed, rather than if one could be achieved.

Conclusion. The results are shown in figure 7 where we have plotted the value for the perfect run, and the mean value for the noisy runs. As expected, including noise at the levels of the NQIT device leads to an outcome probability that is between the ideal and the totally random output. However in most cases the noise that we include leads to a result within one standard deviation of the uniform distribution, or greater than one standard deviation from the perfect run. Referring to our figures of merit, we regard this to be a sign that the scheme is unsatisfactory for demonstrating quantum-advantage with NQIT noise at its current levels.

Zoom In Zoom Out Reset image size

In section 4.3 we use the simulator as a tool to investigate which of the aspects of our noise model are the main sources of this failure. Our intention is to direct subsequent experimental and theoretical research towards diminishing this source, potentially leading to a quicker implementation of quantum-advantage experiments.

To form a complete picture, and to benchmark the device's performance when implementing these problems, we must compare our numerical experiments with actual experiments. This work concerns only numerical experiments, while in the future we plan to collaborate with experimental groups to provide these benchmarks.

The second numerical experiment we perform takes the general IQP-MBQC of section 3.1.3 and imposes constraints equivalent to the architecture of NQIT. We consider the case where each ion-trap has multiple logical qubits, as discussed in section 3.2. Moreover, we restrict to IQP instances involving gates acting on qubits belonging to neighbouring ion-traps so as to lower the circuit depth as much as possible.

In principle different gates of an X-program may act on any subset of qubits, or in the MBQC model, the gate qubits may be entangled with any subset of the application qubits. This is not realistically achieved in the NQIT setting, where qubits belonging in different ion-traps cannot be connected arbitrarily with qubits of other ion-traps. Since NQIT admits universal quantum computation, one could achieve arbitrary connectivity by using swaps between the qubits. However, by doing these swaps the advantage of smaller waiting times offered by IQP is destroyed. We will thus impose conditions on the connectivity, limiting the class of problems we use.

We have assumed that each ion-trap has $K=20$ physical qubits, of which 10 are dedicated to entanglement distillation, leaving $K^{\prime} =10$ for use in computation. As discussed in section 3.3, this allows us to fix the noise of two-qubits gates to be constant, whether it involves qubits in the same or neighbouring ion-traps. This does not apply to the waiting time, and thus decoherence, which is greater in the case of gates involving qubits in different ion-traps.

We will choose the minimum links between different ion-traps (while maintaining full connectivity within each trap). This means a 1 dimensional configuration of ion-traps¹⁹ . This, in itself, might not be a big restriction, since even considering two-qubit gates that act on nearest neighbour qubits only, as shown by theorem 3.3, is still believed to be a hard problem. However, this configuration, while it is not 1 dimensional as far as the qubits are concerned, is still likely to admit a classical efficient simulation based on tensor networks and matrix product states [110]. Since our purpose in this section is to illustrate how to implement architecture constraints, the issue of classical hardness in comparison to the best classical methods, is not crucial. Indeed it is likely that reasonable predictions can be made about the impact of noise on the 2 dimensional architecture designed by NQIT, outputs from which are less likely to be reproducible on a classical computer, using results from these 1 dimensional simulations. In contrast, in the first numerical experiment, there is a complexity-theoretic proof of hardness.

In IQP-MBQC, applying gates between application qubits corresponds to entangling them with the same gate qubit. In the case that the application qubits belong to different ion-traps, the gate is applied using teleportation, with the help of entanglement links distilled between neighbouring ion-traps. Protocol 2 shows how to achieve this using only one entanglement link between the two ion-traps. Distiling entanglement between multiple traps takes a longer time, which is why we restricted our attention to X-programs that involve gates with qubits in at most two ion-traps.

Protocol 2. This algorithm constructs part of the resource state for a given gate qubit g in trap 1 according to its corresponding row p of the X-program ${\bf{Q}}$. Q₁ is the set of all qubits in cell 1 with $g,{l}_{1}\in {Q}_{1}$. Analogously, $c,{l}_{2}\in {Q}_{2}$. c is the qubit that will eventually be used for measurement after g's value is teleported there.

1: function EntangleTwoTraps(p, g, c, l1, l2, Q1, Q2)

2:    for all $q\in {Q}_{1}:p(q)=1$ do

3:      ${CZ}$ (g, q)

4:    end for

5:    ${CZ}$ (g, l1)

6:   Distil a Bell pair between l1 and l2

7:   Bell measurement on (g, l1) which teleports g to l2

8:   SWAP (c, l2)

9:   for all $q\in {Q}_{2}:p(q)=1$ do

10:     ${CZ}$ (c, q)

11:   end for

12: end function

In this setting, we have each ion-trap being connected by entanglement links to two neighbouring ion-traps. Each ion-trap has one gate qubit (g in Protocol 2) and one qubit reserved to receive the gate qubit coming from it's neighbour (c in Protocol 2). This leaves 8 application qubits. This entanglement structure can be achieved in two time-steps. First, all ion-traps at odd positions use their entanglement links to teleport the qubit required using Protocol 2. This is repeated for all even positions. This two-step process is shown schematically in equation (6).

With these restrictions X-programs can be mapped to NQIT's architecture. An example of an MBQC graph for such restricted instances is given in figure 8.

Zoom In Zoom Out Reset image size

Simulations. A full description of the simulation procedure can be seen in appendix B.1.2. In summary, we let each gate qubit act on a random subset of the application qubits in its own ion-trap before, after being teleported, acting on a random subset of the qubits in the next ion-trap. We performed 20 trial, each involving a randomly generated circuit of the form described above, along with a random output string. Each trial has one noisy and one perfect run. A perfect run involves simulating the perfect circuit several times and calculating the mean probability of measuring the selected output string. A noisy run is equivalent but with a random noisy instance of the circuit.

In this case, at their largest, we simulate significantly more qubits than in the previous and following sections. The largest circuit we simulate has $12\times 8$ qubits but still only 10 T gates on average. This is because we have limited the probability that a T gate will be required, which corresponds, as discussed in appendix B.1.2, to limiting the probability of creating connections between the gate and application qubits. As the computation time grows exponentially with the number of T gates, and polynomially in the number of qubits, we can afford this increase in the qubit count.

Figure of merit. In this case, as we expect that the architectural restrictions used will make a demonstration of quantum-advantage using this scheme unlikely, we will not consider the figures of merit as in section 4.2.1. Instead we again consider the coefficient of determination as in section 4.1 to establish the impact of noise models more broadly. Here the model outputs m_i are the probability amplitudes from the noisy run, while the target outputs d_i are those from the perfect run.

Conclusion. We compared the two means of each run to calculate the coefficient of determination. In the case of the maximum system (12 ion-traps, with 8 application qubits each) we noticed that, with the existing level of noise, the results corrupt fully the output leading to ${R}^{2}\approx 0$. We then ran similar experiments for smaller instances. Lowering the number of qubits, we observed that the R² value was increasing but still remained extremely low with NQIT noise level. Decreasing the size yielded the following results (a × b means a ion-traps with b application qubits per trap):

$$\begin{eqnarray*}&&\left.\begin{array}{c}a\times b\\ {R}^{2}\end{array}\right|\begin{array}{cccc}12\times 8 & 9\times 8 & 4\times 8 & 4\times 2\\ 0.008\,6 & 0.023\,7 & 0.033\,3 & 0.556\,1.\end{array}\end{eqnarray*}$$

These R² values, far below one, indicate that even for small system sizes, the noise is too high and there is little correlation between the perfect and noisy runs. For this reason, and because theoretical results about quantum-advantage in this case are not as strong, in the subsequent section where we examine the effects of varying noise, we restricted attention to the numerical experiment of section 4.1 only and do not proceed to Part 3 of the numerical experiments in this case.

At the coarsest level of detail, we group time-based noise (depolarising and dephasing) together, and operation-based noise (preparation, measurement, single and two qubit gates, including the noise during distillation) together. In each run we eliminate either the time-based noise or operation-based noise, while keeping the other at the same level as in NQIT's device. Result for the behaviour of outputs with far from uniform probability in the ideal output distribution can be seen in figure 9.

Zoom In Zoom Out Reset image size

We can see that the largest contribution to the corruption of the output appears to be from the time-based noise. When we were exploring candidates for demonstrating quantum-advantage, we mentioned that time based noise is frequently a major issue. This motivated us to consider IQP and here our results justify this choice.

With reference to our figures of merit, including only time-based noise almost always brings the output probability of the bit string in noisy runs to within one standard deviation of the uniform value, or greater than one standard deviation away from the perfect run amplitude value. As such we conclude that it is a significant obstacle to demonstrating quantum-advantage. On the other hand, as the randomly selected bit string amplitude, when only gate based noise is considered, is in all but one case within one standard deviation of the perfect run, and further than one standard deviation from the uniform distribution value, we do not immediately conclude that it is a significant obstacle.

Below the reader can see values for a proxy for the ℓ₁-norm distance between the ideal and noisy distributions for the noise levels discussed above, calculated as follows. Here a trial consists of an ideal run, measuring the probability of a random output of a random 2D-DQS circuit of the form discussed in section 3.1.4, and 20 noisy runs for each noise type, considering noisy versions of the ideal circuit. The average difference between the noisy and ideal runs within each trial are themselves averaged to give a proxy for the ℓ₁-norm distance, once scaled by the uniform distribution. Each run is itself the average of 20 simulations of the same circuit.

A similar pattern is seen in this data as was identified in the study of single outputs; namely that the largest contribution to the deviation of the noisy distribution from the ideal is a result of the time based noise.

As discussed in section 4 our analysis of both far from uniform outputs and the ℓ₁-norm distance lead us to regard a system with reduced time-based noise as relatively more likely to demonstrate quantum advantage than a system with reduced gate-based noise. In this case, removing the time based noise results in a value below the $\tfrac{1}{22}$ specified in theorem 3.4 suggesting that a demonstration of quantum-advantage may be possible here. However, we hope to identify the main source of error more precisely, and as such we continue to explore the reduction of time-based noise.

We now look more closely at the time-based noise and consider separately the contribution from dephasing noise and from depolarising noise. The results for outputs with far from uniform probability in the ideal output distribution are seen in figure 10.

Zoom In Zoom Out Reset image size

In this case, the amplitudes produced by runs considering only dephasing noise are always either within one standard deviation of the uniform distribution, or greater than one standard deviation from the perfect run. By comparison the runs considering only depolarising errors are always within one standard deviation of the the perfect run, and greater than one standard deviation from the uniform distribution output.

Below the reader will again find the same proxy for the ℓ₁-norm distance between the ideal and noisy distributions as discussed in section 4.3.1, but for the noise levels considered in this section. A similar pattern is seen in this data as was identified when considering the accuracy and far from uniformity figure of merit above; namely that the largest contribution to the deviation of the noisy distribution from the ideal is a result of the dephasing noise.

As as a result of the analysis of these two figures of merit, we identify dephasing error as a relatively larger obstacle to a demonstration of quantum-advantage than depolarising noise.

Having identified the main obstacle to a demonstration of quantum-advantage to be dephasing errors, we examine the effect that reducing this type of noise would have. Concretely, one could introduce a phase-flip code²⁰ [111]. Recall that in the numerical experiments of section 4.1, we only used a single qubit from each ion-trap. This means that we could use three qubits from the ion-trap to implement one round of phase-flip code, which would reduce the dephasing noise. By using such a simple phase-flip code we obtained an effective improved dephasing rate of $\approx 2.3\times {10}^{-4}$ s⁻¹ from the one of NQIT noise-level $\approx 7.2\times {10}^{-3}$ s⁻¹. The results for outputs with far from uniform probabilities are found in figure 11.

Zoom In Zoom Out Reset image size

In this case, roughly half of the runs considering the error corrected dephasing pass our test that the probabilities should be at least within one standard deviation of the perfect run, and greater than one standard deviation of the uniform distribution. This demonstrates partial improvement while being inconclusive as a demonstration of the potential for quantum advantage. In this case an analysis of the ℓ₁-norm distance is particularly valuable.

The readers will find the data required for such an analysis below. In this case, as in the case of the previous figure of merit, a large improvement can be achieved by utilising a simple repetition code. However this improvement might not be as significant as one might expect having seen the results of figure 11 with the ℓ₁-norm distance still being significantly far from the $\tfrac{1}{22}$ value required by theorem 3.4. Indeed even without dephasing noise the ℓ₁-norm distance its too high to expect a demonstration of quantum-advantage. As such we expect both improved error correction codes and error correction applied to other noise channels are required for a demonstration of quantum-advantage.

This is a partial improvement relative to the uncorrected results, and so we find a demonstration of quantum-advantage using this error correction scheme as more likely than in the uncorrected case. However, further improvements are required for such a demonstration.

More generally than testing a single error correction code, we can understand how the likelihood of a demonstration of quantum-advantage is affected with a continuously varying noise parameter. Here we will consider dephasing errors, which we have identified as the most damaging form of error. This continuous variation corresponds to, for example, reductions in the gate application time, improvements in the compilation methods or the improved storage of quantum states. The results of this experiment are shown in figure 12.

Zoom In Zoom Out Reset image size

While figure 12 appears to demonstrate the continuous improvement which can be achieved by reducing the dephasing error, it seems that it cannot be said that the amplitudes are regularly within one standard deviation of the perfect run until the dephasing rate is reduced to 0. We do however see that, with regards to our accurate and far from uniform condition, a demonstration of quantum supremacy does become continuously more likely as the dephasing error rate is reduced.

This fact is reinforced by figure 13 which shows the average difference between the perfect and noisy runs for each of the values of dephasing error rate. We can use this as a proxy measure for the ℓ₁-norm distance, as discussed in the experimental design methodology introduced in section 4, and as was done earlier in section 4.3. In this case we can say that an experiment has a reasonable chance of demonstrating quantum-advantage if we can be convinced that the ℓ₁-norm distance between the noisy and perfect implementations is bounded by $\tfrac{1}{22}$ which is demanded by the hardness result for the 2D-DQS algorithm as seen in theorem 3.4. As we do not have access to the full characterisation of the probability distributions, here we will approximate the ℓ₁-norm distance by taking the average difference and proposing that it is representative of the full distribution by scaling it by the uniform distribution.

Zoom In Zoom Out Reset image size

We see that even in the case of 0 dephasing error, the ℓ₁-norm distance is not brought within the $\tfrac{1}{22}$ value. Instead The average difference in that case is approximately 0.155 which is significantly higher. However, by our figure of merit, a demonstration of quantum-advantage is made continuously more likely by this fall in dephasing error, showing the advantage in endeavouring to achieve such a fall.

An alternate proxy measure for the ℓ₁-norm distance is to explore the differences between the noisy and perfect amplitudes for a selection of different output bit strings of the same circuit. In figure 14, every trial considers the same 2D-DQS circuit, but measures the probability amplitude of a different output bit string.

Zoom In Zoom Out Reset image size

Once again, this plot can be examined further by directly studying the average difference between the noisy and perfect runs. This proxy measure for the ℓ₁-norm distance is plotted in figure 15 but once again the value of 0.135 is more than twice the $\tfrac{1}{22}$ which is demanded by the hardness result for the 2D-DQS algorithm as seen in theorem 3.4.

Zoom In Zoom Out Reset image size

In conclusion, while it seems that reducing dephasing error alone will not be enough to bring a demonstration of quantum-advantage using this scheme within reach, we have seen that utilising a simple 3 qubit correction code would result in a significant improvement on the noise levels. As such we recommend that this error correction technique is used in conjunction with other techniques, correcting for other error types.

We expect, however, that as the system size grows the ℓ₁-norm distance between the perfect and noisy circuits will grow as the noise modelled is constant for each gate and qubit. This would push a demonstration of quantum-advantage further away, which is consistent with the theoretical results in [35]. There the authors demonstrate that samples can be efficiently drawn by a classical computer from a distribution produced by an IQP circuits subject to independent depolarising noise on each qubit at the end of the circuit. In that case, however, they show that error correction can be used to recover classical impossibility, if one allows for more complex connectivity, or several rounds of swap gates. While we have restricted the connectivity and circuit depth in our case, there may be gains to be made by removing these restrictions.

Generating random unrestricted IQP instances is equivalent to randomly populating ${\bf{Q}}$ with zeros and ones. The description in appendix A.1 of how to convert a given X-program ${\bf{Q}}$ to a particular circuit lets us control the T-gates count t. We saw that every individual exponential (row in ${\bf{Q}}$) corresponds exactly to $t=1$, and the number of application qubits has no effect on t. We want T-gate counts of no more than 20 in order to achieve feasible run-times.

One trial consists of generating a random IQP instance, obtaining the true probability of measuring the ${\left|0\right\rangle }^{n}$ using brute-force, and solving them with the simulator of [74] 20 times. Each instance is created by randomly populating with binary values a matrix ${\bf{Q}}$ of randomly picked dimensions in $[5,15]\times [5,12]$. This corresponds to $n\in [5,12]$ and $t\in [5,15]$ where the complexity in the brute-force case is determined by n, and in the simulator's, by t.

The experiment consists of 20 trials, with the mean of the simulator output in each trial compared to the brute force case to give the coefficient of determination.

We again generate random IQP-MBQC circuits, but under the restrictions described in section 4.2.2. Rather than a full matrix, ${\bf{Q}}$, it is now sufficient for each gate qubit, g_i, in an ion trap, i, to have corresponding bit strings, i⁰ and i¹, indicating the entanglement patterns between itself and qubits in it and its neighbouring ion trap.

Details of the circuit simulated can be seen in Protocol 4. Once the circuit is simulated, we calculate the probability that an NQIT implementation would measure a random bit string b. One noisy run consists of simulating the circuit produced from Protocol 4, using fixed ${i}^{0},{i}^{1},b$, 20 times to calculate the mean and standard deviation. Then a new tuple ${i}^{0},{i}^{1},b$ is generated and the process is repeated for the next trial. A perfect run is equivalent but with the noise values set to 0, with the perfect and noisy pair forming one trial. In total the experiment consists of 20 trials.

Protocol 4. Code producing a noisy IQP-MBQC circuit, to be implemented by the simulator, as discussed in section 4.2.2. We use i to index the ion traps, and to represent the set of $K^{\prime} -2$ available application qubits which each trap contains ($K^{\prime}$ minus 1 qubit c_i to receive the gate qubit from it's neighbour, minus one gate qubit g_i).

Input: For every ion trap, i, two strings, i0, i1. Bit string b.

Output: Noisy circuit.

1: for all $q\in$ qubits

2:    Initialise(q)        ▷Recall, initialisation is in the $\left|+\right\rangle$ state

3:    PreparationNoise(q)

4: end for

5:

6: for all $i\in$ ion traps, except the last do

7:    for all $q\in i$ do

8:      if ${i}_{q}^{0}=1$ then

9:         Act ${CZ}\left({g}_{i},q\right)$

10:        TwoQubitNoise(gi, q)

11:        TimeBasedNoise(TimeInTrapOperation)

12:      end if

13:    end for

14: end for

15:

16: for all $i\in$ ion traps, except the last, such that i is even do

17:    Swap(gi, ${c}_{i+1}$)        ▷Move gate qubits to neighbouring ion trap

18: end for

19: TimeBasedNoise(TimeLinkingOperation + TimeMeasurement)

20:

21: for all $i\in$ ion traps, except the last, such that i is odd do

22:    Swap(gi, ${c}_{i+1}$)

23: end for

24: TimeBasedNoise(TimeLinkingOperation + TimeMeasurement)

25:

26: for all $i\in$ ion traps, except the first do

27:    for all $q\in i$ do

28:      if ${\left(i-1\right)}_{q}^{1}=1$ then

29:        Act ${CZ}\left({g}_{i-1},q\right)$

30:        TwoQubitNoise(${g}_{i-1}$, q)

31:        TimeBasedNoise(TimeInTrapOperation)

32:      end if

33:    end for

34: end for

35: for all $i\in$ ion traps, except the first do

36:    Act $H\left({g}_{i-1}\right)$

37:    SingleQubitNoise(${g}_{i-1}$)

38: end for

39: TimeBasedNoise(TimeInTrapOperation)

40: for all $i\in$ ion traps, except the first do

41:    Act $T\left({g}_{i-1}\right)$

42:    SingleQubitNoise(${g}_{i-1}$)

43: end for

44: TimeBasedNoise(TimeInTrapOperation)

45: for all $i\in$ ion traps, except the first do

46:    Act $X\left({g}_{i-1}\right)$

47:   SingleQubitNoise(${g}_{i-1}$)

48: end for

49: TimeBasedNoise(TimeInTrapOperation)

50:

51: for all $q\in$ qubits do

52:    Act $H\left(q\right)$

53:    SingleQubitNoise(q)

54: end for

55: TimeBasedNoise(TimeInTrapOperation)

56:

57: for all $i\in$ ion traps, except the last do         ▷CNOT seen at end of figure A1

58:    for all $q\in i$ do

59:      if ${i}_{q}^{0}=1$ then

60:        Act ${CX}\left({g}_{i},q\right)$

61:      end if

62:    end for

63: end for

64:

65: for all $i\in$ ion traps, except the first do

66:    for all $q\in i$ do

67:      if ${\left(i-1\right)}_{q}^{1}=1$ then

68:        Act ${CX}\left({g}_{i-1},q\right)$

69:      end if

70:    end for

71: end for  72:

73: for all $i\in$ ion traps do

74:    for all $q\in i$ and ${g}_{i-1}$ for all but the first ion trap do

75:      MeasurementNoise(q)

76:    end for

77: end for

78: Measure(b)     ▷Give the probability of measuring b in the Computational basis

Notice that we are being pessimistic in Protocol 4 by assuming that there is no parallelism in the gate applications. As such we apply time based noise after each gate. We have also simplified the operation of swapping to a single operation, rather than a protocol as seen in Protocol 2. This reduces the simulation time while roughly maintaining the noise impact, as the time based noise should dominate here.