Introduction: The complications of a noisy quantum world . 1
Random quantum circuits and Random Circuit Sampling
The white-noise distribution is a mixture of the ideal distribution pideal and the uniform distribution punif, for which each outcome has an equal probability 2−n of being obtained (complete white noise). Numerical evidence in favor of the white noise assumption for random quantum circuits under a local noise model was provided in Ref.
Progress on hardness of simulation for random circuits in prior
Essentially, it has been shown that for any worst case one can generate a set of random cases and then derive the output probability of the worst case from the output probabilities of the random cases. Moreover, this derivation process has been shown to be robust to a small amount of error: as long as the mean-case output probabilities can be derived precisely enough for the argument to additional precision at most e−Ω (nlog the worst-case output probability is calculated to go through.
Quantum computational supremacy on noisy devices
In fact, even if the white noise distribution is a good approximation for the output of the device, we expect there to be a small total variation distance error between the noisy distribution sampled. This illustrates how the white noise assumption is important in justifying the use of the linear cross entropy to measure noise in the experiment.
Overview of results
Along the way to prove the white noise assumption, we also give tight bounds on the linear standard decomposition of the cross-entropy (which depends only on the mean infidelity) and the rate of convergence of the output distribution to the uniform distribution (which depends only on on unitarity). We find that the white noise assumption holds only when the noise channel is mostly incoherent.
Outlook for simulation complexity of random quantum circuits 18
In the general case, SEBD follows O(r) columns simultaneously where r is the radius of the light cone corresponding to the circuit. Suppose we are interested in the quasi-entropy S˜k(A) = (Fk,A−Fk,∅)/(k−1) of the output distribution in this region.
The statistical mechanics method for random quantum circuits 21
Statistical mechanics, partition functions, and the Ising model . 22
The coefficients of the above four terms are transformed into edge weights for the partition function. This version of the stat mech method is derived in a standalone manner in Chapter 4.
Past and future of the stat mech method
Since each of the s gates in the circuit diagram acts on two indices, it must hold that P. We let eH,j,t be the location of the jth domain wall at time in the trajectory H.
Efficient simulation of shallow 2D random quantum circuits 32
Overview of contributions
Since these results naturally give worst-case robustness, they do not clearly imply that simulating the random circuit must be difficult. The uniform random circuit family for which we gather the most evidence for classical simulability is related to the depth-3 "brick architecture".
Simulation by reduction to 1D dynamics
This will usually reduce the bond dimension of the MPS, depicted in the cartoon, by a reduction in the thickness of the lines between tensors. For an output string x∈ [q]n, let DC(x) denote the probability that the circuit will output x after measurement.
Rigorous analysis of SEBD for the “extended brickwork archi-
Finally, in the toy model, we assume for simplicity that the EHR pairs move cyclically. Despite these simplifications, we believe that this model is qualitatively accurate in the territorial phase. We first note that Brickwork(L, r(L), v(L)) supports universal MBQC in the sense that a specific choice of gates can create a resource state that is universal to MBQC.
Our goal is to prove that SEBD can effectively approximate the expanded masonry architecture in the average case for the choice of expansion parameters for which the above hardness results hold. In the effective 1D dynamics, therefore, no 2-qubit maps are used, and therefore the bond dimension of the associated MPS cannot increase during these stretches. A more precise account of the efficiency of SEBD for this architecture is given in the lemma below, the proof of which can be found in Appendix 3.E.
Numerical results
The difference between maximum grid sizes achieved for the two architectures is a consequence of the fact that the two have very different effective 1D dynamics. In particular, the entanglement of the effective dynamics for the brickwork architecture saturates to a significantly smaller value than that of the group state architecture. And even more directly relevant to prospects of. fast simulation, the typical spectrum of Schmidt values over some cut of the effective 1D dynamics for the brickwork architecture decays much faster than that of the 1D dynamics for CHR.
For this reason, the slower decaying eigenvalue spectrum of CHR was significantly more expensive to the algorithm's running time. For further proof of effectiveness, we study the functional form of the entanglement spectra for the effective 1D dynamics. In the case of the masonry architecture, we are also able to provide independent analytical evidence (Section 3.6.6) that this is the case by showing the "quasi-entropy" S˜2 of the 1D process is in the field-law phase.
Analytical evidence for conjectures from statistical mechanics . 66
Straight carries weight equal to qC where C is the number of cycles in the product of the two adjacent permutations. Note that the application of the dephasing channel is not described in the formalism we discussed previously, but is easily incorporated. We first obtain exact, closed-form results in the zero-temperature limit of the stat mech model, which corresponds to the q→ ∞ limit.
We first briefly review their technique before discussing a limitation on the robustness of the polynomial interpolation scheme. It is also proportional to the second moment (and thus related to the variance) of the output probability of a randomly chosen bit string. This lemma is a more precise and generalized version of the recursive computation of Q(x) in Section 4.5 in the main text.
Here, the peak of the probability measure in the initial configuration Λb starts around the Hamming weight x=n/(q+ 1).
Anti-concentration of random quantum circuits in logarith-
Motivation
In this chapter we focus on another property of random quantum circuits called anti-concentration. On the one hand, anti-concentration is a necessary ingredient in most formal hardness arguments for RQC simulation. In this chapter, we prove sharp bounds on the number of gates needed for anti-concentration in two RQC architectures.
For 1D RQC, we confirm an O(log(n)) upper bound for the counterconcentration depth in Ref. We show that, for qubits (local dimension q = 2), a 5nlog(n)/6 gate is necessary and sufficient (up to subroutine corrections) to achieve counterconcentration, confirming the conjecture in Ref. First, we show that counterconcentration is generally achieved much faster than the 2-design property.
Setup and definition of anti-concentration
This comes in sharp contrast to the situation for unitary designs, where the scale of the size needed with n is very dependent on the architecture. Second, the fact that we can prove tight upper and lower bounds attests to the power of the stat mech method and suggests that it may be similarly useful in other situations. Most of the probability mass is concentrated on a few measurement outcomes, with the rest of the outcomes being assigned a very small amount of mass, resulting in a large frequency of circuit cases for which pU(0n) is close to 0.
A uniform distribution, where each output is assigned a q−n proportion of the total probability mass, is a finite anti-concentrated distribution because the mass is perfectly uniformly distributed, but the distribution is still said to be anti-concentrated as long as the average age deviations from the uniform are not greater than O( q−n). The collision probability gives the probability that the results of measurements from two independent copies of the circuit are the same. If desired, Markov's inequality can then be used to limit the fraction of a randomly selected U whose collision probability is greater than some constant multiple Z .
Overview of contributions
Similarly, we show an upper bound for the entire graph architecture, where each gate acts on an arbitrary pair of qudits without considering their spatial proximity. A circuit diagram of size s randomly selected from the entire graph architecture will have a high probability of having a depth of at most O(slog(n)/n) [129], meaning that O(log(n)2)depth is usually sufficient for anti-concentration in full graph architecture. This result is weaker than our specific results for the 1D and full-graph architectures, and we suspect that it can be amplified to Θ(log(n)) gates per qudit.
In particular, we show a general lower bound as well as specific lower bounds for the 1D and full graph architecture. We note that, for q ≥ 5, our results have the counter-intuitive implication that the 1D architecture converges faster than the full graph architecture, even though it is geometrically local. We suspect that a parallelized version of the full graph architecture would counterbalance with a slightly better constant than the 1D architecture.
Related work and implications
As q increases, anti-concentration becomes arbitrarily fast for the 1D architecture (the coefficient of thenlog(n) decreases like1/log(q)). Interestingly, TPEs also bound the collision probability to additive error, but again the error must be exponentially small to achieve anti-concentration. The primary role counterconcentration plays is to change a small additive difference|pU(x)−p0U(x)|for mostex to a small relative differencer(x)for most x, where.
Even if there is anti-concentration, more is needed to demonstrate the rigor of approximately simulating RQCs. Our work places tight limits on the number of gates required to enforce anti-concentration across multiple RQC architectures, which limits when these hardness arguments have the potential to work. Our finding that the number of gates per qudit required for anti-concentration grows only logarithmically in the 1D and full-graph architectures implies that RQC-based quantum computational supremacy could be achievable at a smaller circuit depth than previously thought.
Summary of method and intuition for logarithmic convergence 132
In the second example, the trajectory reaches one of the fixed point configurations where all n values agree; this is not the case in the first example. Trajectories that reach a fixed point quickly generally have larger weights and make the largest contribution to the collision probability. As the size of the circuit increases, more of the trajectories approach the fixed point and Z approaches ZH.
We use the framework of the biased random walk, given by the expression for Z in Eq. Suppose we fix a domain wall configuration g(0) for the initial time step at the beginning of the circuit with k domain walls. Meanwhile, the sum of the weights of all domain wall trajectories in GO approaches ZH(q+ 1)n/2 from below as the depth increases.
In the following sections, we complete the proofs for the upper and lower bounds of the complete graph architecture formally defined in Definition 4.1. In the rest of the proof, we will use the notation v˜= min(v, n−v) for any integer v.
Approximation of noisy random quantum circuits as ideal
Motivation
Noise model and random quantum circuits
Overview of contributions
Related work and implications
Summary of method and intuition
Outlook
