Chapter IV: Anti-concentration of random quantum circuits in logarith-
4.2 Setup and definition of anti-concentration
In the previous chapter, we discussed various constant-depth 2D RQC architectures, which were families of quantum circuit diagrams indexed by the circuit size n, where the actual gates in the diagram were always chosen randomly from the Haar measure. In this chapter, we extend this concept to deeper circuits and define an RQC architecture as simply an instruction set on how to draw a circuit diagram given both the number of qudits n (each with local Hilbert space dimension q) and the size s of the circuit. The two architectures we consider specifically in this chapter are the 1D architecture (with periodic boundary conditions), where qudits are arranged in a ring and alternating layers of nearest-neighbor two-qudit gates are performed, and the complete-graph architecture, where each two-qudit gate is chosen uniformly at random among alln(n−1)/2possible qudit pairs. Formal definitions of these architectures appear in Appendix 4.A. Note that in the formal analysis, we also include a layer ofnsingle-qudit gates at the beginning of the circuit, which are not counted toward the circuit size s. These gates might be regarded as fixing the local basis for the initial input state.
The associated RQC ensemble for an RQC architecture is formed by fol- lowing this instruction set to obtain a circuit diagram, and then choosing the value of each gate in the diagram independently from the Haar measure.
Choices for each gate determines an overall qn×qn unitary U that the cir- cuit implements. Each instance U is associated with an output probability
frequency
output probabilitypU(x)
Uniform Z =q−n
frequency
output probabilitypU(x)
Anti-concentrated Z≈c q−n
frequency
output probabilitypU(x)
Not anti-concentrated Z ≥encq−n
Figure 4.1: A caricature of anti-concentration. For the uniform distribution, which is completely anti-concentrated, all qn outcomes are allocated probabil- ity mass q−n (dashed blue line) and the collision probability is Z = q−n. For globally Haar-random unitaries, the output probabilities are on average q−n but have some non-zero variance, and the collision probability is Z ≈ 2q−n. WheneverZ ≈cq−nfor somecindependent ofn, we call the distribution anti- concentrated. For low-depth RQCs, the mean output probability is q−n, but the variance is much larger, and the collision probability is much larger than q−n. Most of the probability mass is concentrated onto a few measurement outcomes, with the remainder of the outcomes being assigned a very small amount of mass, leading to the frequency of circuit instances for which pU(0n) is close to 0 to be large.
distribution pU over qn possible computational basis measurement outcomes x ∈ [q]n, (where [q] = {0,1, . . . , q−1}): pU(x) = |hx|U|0ni|2. Note that this distribution is referred to by the notationpideal inChapter 1 and Chapter 5.
Anti-concentration tries to capture the notion that the probability mass in pU is well spread out over all the outcomes. The uniform distribution, where each output is allocated q−n fraction of the total probability mass, is the ultimate anti-concentrated distribution because the mass is exactly equally spread, but we say a distribution is still anti-concentrated as long as the av- erage fluctuations from uniform are no larger than O(q−n). This definition is captured precisely by the collision probability, which is P
xpU(x)2. The col- lision probability gives the probability that measurement outcomes from two independent copies of the circuit are identical. It is also proportional to the second moment (and thus is related to the variance) of the output probability of a randomly chosen bit string. If pU is the uniform distribution, then the collision probability is q−n, its minimal possible value. For an RQC architec- ture at a specified qubit number n and circuit sizes, we consider the collision probability averaged over the randomly chosen circuit instancesU, defined by the expression
Z =E
U
X
x∈[q]n
pU(x)2
=qnE
U
pU(0n)2
, (4.1)
where the second equality holds because by symmetry, each of the qn terms in the sum yields the same number under expectation as long as at least one Haar-random gate acts on each qudit.
We say an RQC architecture withnqudits andsgates is anti-concentrated if there is a constant α (independent of n) with 0 < α ≤ 1 for which Z ≤ α−1q−n, i.e. that the collision probability is only a constant factor larger than its minimal value. In particular, our theorem statements roughly correspond to the choice α= 1/4, but other choices of α would yield the same results up to leading order. If desired, Markov’s inequality can then be used to bound the fraction of the randomly chosen U whose collision probability is larger than some constant multiple of Z. Moving forward, for convenience, when we say collision probability we will mean the average collision probability Z.
Very shallow circuit architectures are not anti-concentrated: there are ex- pected to be some output probabilitiesxfor whichpU(x)is exponentially larger than the mean of q−n. As the circuit gets deeper, we expect the probability distribution to become closer to uniform, but even at infinite depth, when the circuit unitaryU becomes a globally Haar-random qn×qn unitary, the output distribution still does not become completely uniform. In this case, the output distribution will typically follow a Porter-Thomas distribution1, andZ can be
1In the Porter-Thomas distribution, the frequency at which pU(x) =pis proportional toexp(−p/q−n), illustrated roughly in the middle diagram ofFigure 4.1.
exactly computed as
s→∞lim Z =ZH = 2
qn+ 1, (4.2)
roughly twice as large as the minimal value ofq−n associated with the uniform distribution. This statement is proved using the techniques described later.
Refer to Figure 4.1 for a graphical illustration of these cases.
While one could capture the notion of anti-concentration with a differ- ent definition, the definition we choose is useful and relevant because it has concrete ramifications in all of the previously mentioned applications of anti- concentration. For example, one implication of our definition (by application of the Paley-Zygmund inequality) is that ifZ ≤α−1q−n, then for any0≤β≤1, PrU[pU(x)≥βq−n]≥α(1−β)2, (4.3) meaning that for at least a constant fraction of the circuit instances the prob- ability of a given measurement outcome x is at least a constant fraction β of the mean measurement probability q−n. This sort of inequality is the relevant one for turning good additive approximations into good multiplica- tive approximations (with reasonable probability), employed in, for example, Refs. [4,19,22,134–138] in order to argue that it is hard to classically sample output distributions for a large fraction of instances up to small total varia- tion distance error (for more details, see Section 4.4.2). In fact, equations like Eq. (4.3) are sometimes taken to be the definition of anti-concentration [136], which is a weaker definition than ours since, in principle, it can hold even in cases where our definition does not.