Chapter II: The statistical mechanics method for random quantum circuits 21
2.2 Statistical mechanics, partition functions, and the Ising model . 22
cal systems like temperature, energy, and entropy to their microscopic descrip- tions. The core idea is that for a certain macroscopic state of the system, there
the transformation enacted by the circuit is not strictly unitary, but the stat mech method is still useful.
is some ensemble of possible corresponding microstates. After all, knowing the temperature, pressure, and volume of gas in a room hardly tells you the exact location of all the particles, but it does tell you something.
The Ising model is an illustrative example of classical statistical mechanics in action, and one that we will come back to in the context of random quantum circuits. In the Ising model, we have a system of m spin-1/2 particles, each with two internal states, so the system microstates are labeled by a choice of σi = ±1 for each i = 1, . . . , m, collectively denoted by σ. The energy of a microstate is given by
H(σ) =−
m
X
i=1 m
X
j=i+1
Jijσiσj, (2.1)
where the matrix Jij encodes the interaction strengths between particle i and particle j. The pairs of particles hiji (with i < j) for which Jij 6= 0 form the edges of the interaction graph for the model; often we restrict this interaction graph to be spatially local, for example on a 1D or 2D lattice. One macroscopic quantity of interest is the magnetizationM =P
iσi, which represents the total magnetic moment of the system as a whole.
If the system is in thermal equilibrium at some temperature T, then in- troductory statistical mechanics dictates that the system is in the canonical ensemble and the probability that the system is in microstate σ is
Pr[σ] = 1 Z exp
−H(σ) kBT
, (2.2)
where kB is the Boltzmann constant and Z is the partition function, given as follows:
Z =X
σ
exp
−H(σ) kBT
=X
σ m
Y
i=1 m
Y
j=i+1
exp
Jijσiσj kBT
(2.3)
=X
σ
Y
hiji
weighthiji(σ), (2.4)
where the sum over hiji denotes a sum over edges of the interaction graph.
The definition
weighthiji(σ) = exp
Jijσiσj kBT
(2.5) emphasizes that the partition function is simply a weighted sum over all pos- sible microstates of the m-particle system, where all the weights are positive numbers, and furthermore each weight can be decomposed into a product of edge weights for each edge hiji in the interaction graph; the edge weight for edge hiji only depends on the internal states of particles i and j.
The Ising model is used to understand ferromagnetism in materials as the temperature changes. Suppose all Jij are non-negative so that the minimum
energy microstates are the microstates where all of the particles have the same internal state: either σi = 1 for all i or σi = −1 for all i. In either of these microstates, the system is highly polarized, with the total magnetization M =
±m. At T =∞, all microstates are equally likely, but as T decreases, lower energy states become more probable. In some cases, there is a critical value of T that divides two distinct phases. The high-temperature “paramagnetic”
phase is characterized by disorder and no macroscopic magnetization; that is, microstates drawn from the canonical ensemble typically have small values of |M|. The low-temperature “ferromagnetic” phase is characterized by long- range order and macroscopic magnetization; that is, |M| = Θ(m). This kind of order-disorder thermal phase transition happens for the 2D Ising model, but not for the 1D Ising model, where the paramagnetic phase persists for all T > 0. In Chapter 3, we argue that a similar phase transition occurs in the classical model associated with 2D random quantum circuits. This phase transition is driven not by temperature but rather by features of the quantum circuit—specifically, the depth and the local Hilbert space dimension of the qudits.
2.3 The map from random quantum circuits to classical partition functions
The map from quantum circuits to classical stat mech systems depends on the quantum circuit diagram (i.e., the arrangement of two-qudit gates), the local Hilbert space dimension q of the qudits, and the particular quantity of interest that relates to the kth moment of the random circuits. It is simplest to assume that k = 2 and then generalize to larger k.
For k = 2, we are interested in the expectation value of quantities f(U) where
f(U) = L U⊗2⊗U∗⊗2
(2.6) for some linear functionL. In fact, sinceU is composed of the smaller unitaries U(t) for t= 1,2, . . . , s, the functionL is linear in
U(t)⊗2⊗U(t)∗⊗2 (2.7)
for each t. Since each U(t) is chosen independently at random, we can per- form the expectation value over choice of U(t) individually for each t, which is possible because second moment expectation values over the Haar measure have a closed-form expression. We express this key formula for single-qudit q×q Haar-random unitaries. To so, we first choose any basis {|ii}q−1i=0 for the Hilbert space and define vectors |Ii and |Si, which live in a four-fold tensor
product of the Hilbert space.
|Ii=
q−1
X
i1=0 q−1
X
i2=0
|i1, i2i ⊗ |i1, i2i (2.8)
|Si=
q−1
X
i1=0 q−1
X
i2=0
|i1, i2i ⊗ |i2, i1i. (2.9)
Then, with R
dV denoting integration over the Haar measure, we have the following formula [33]:
Z
dV V⊗2⊗V∗⊗2
= 1
q2−1|IihI|+ 1
q2−1|SihS| − 1
q(q2−1)|IihS| − 1
q(q2−1)|SihI|. (2.10) In the case that V is a two-qudit unitary, we simply send q → q2, |Ii →
|Ii⊗2 and |Si → |Si⊗2 in the formula above.
The appearance of the stat mech partition function can be seen directly from Eq. (2.10): each two-qudit gate U(t) in the circuit diagram gets replaced with a pair of particles, an incoming particle associated with the bras in the equation and an outgoing particle associated with the kets. These particles can be in one of two internal states |Ii⊗2 or|Si⊗2, and the formula is a weighted sum over all possible internal states of those two particles. Let τt denote the internal state of the tth incoming particle and σt denote the internal state of the tth outgoing particle. The coefficients of the four terms above turn into edge weights for the partition function.
U(t)
Figure 2.2: Map from a two-qudit gate in the random circuit diagram to a pair of particles, an incoming (blue) particle and an outgoing (red) particle.
These particles have an interaction given by Eq. (2.11) and depicted by the zigzag line.
By applying the formula to each of the random gates that comprise U, a circuit with s gates turns into a stat mech system with 2s particles, each with two possible internal states, which gives a total of22ssystem microstates.
There is an interaction between the two particles arising from the same unitary t, which is denoted by the edge hti. The edge weight of hti is read off from
Eq. (2.10), with q replaced byq2 since the gates act on two qudits.
weighthti(σ, τ) =
((q4−1)−1 if σt=τt
−q−2(q4−1)−1 if σt6=τt. (2.11) There are also interactions between an outgoing particle from unitary u and an incoming particle from unitary v if the two unitaries act in succession on the same qudit. We denote this edge as huvi. The edge weight ofhuviis given by
weighthuvi(σ, τ) =
(q2 if σv =τu
q if σv 6=τu, (2.12) owing to the fact that hI|Ii = hS|Si = q2, and hI|Si = hS|Ii = q. In the case that measurements or noise channels act in between unitaries u and v, the weight formula would be modified.
The final ingredient to the correspondence is the choice of boundary con- ditions at the beginning and end of the quantum circuit diagram. These will depend on the quantity f that we are trying to compute. For example, in Chapter 4, we computeEU[h0n|⊗4U⊗2⊗U∗⊗2|0ni⊗4]and in that case we have open boundary conditions on both sides. However, in Chapter 3, we will see an instance where more complicated boundary conditions at the end of the circuit are required. For this general discussion, we will stick to looking only at the bulk properties of the system. An example of the map from circuit diagram to interaction graph appears in Figure 2.4.
Together, these observations allow us to write EU[f(U)] =X
σ,τ
Y
hti
weighthti(σ, τ)Y
huvi
weighthuvi(σ, τ), (2.13) mirroring the equation for the partition function in Eq. (2.4).
One looming difference between this partition function and that of the Ising model is the possibility of negative weights, as seen in Eq. (2.11). This is a manifestation of the sign problem, and is problematic for a few reasons.
First of all, it is impossible to interpret the stat mech system as an actual physical system at a real-valued temperature, making connections to conven- tional statistical mechanics less direct. Second, it is possible that the positive terms and negative terms in the sum are both very large but cancel out to yield something small, complicating combinatorial methods that might try to put upper and lower bounds on the quantity by taking the absolute value of the individual terms. Later, we will show how the issue of negative weights can be circumvented in the case of k = 2.
Higher moments
The generalization of the method to moments for k ≥ 3 is straightforward.
In fact, the interaction graph for the stat mech system is independent of k.
However, instead of two internal states, each particle has k! possible internal states, which can each be associated with an element from the symmetric group Sk. These elements are permutations of the indices{1, . . . , k} and can be written in cycle notation; for example, the swap operation for k = 2could be written as (12) instead of S. For ν ∈ Sk, we define
|νi=
q−1
X
i1,...,ik=0
|i1, . . . , iki ⊗ |iν−1(1), . . . , iν−1(k)i (2.14) on 2k copies of the Hilbert space, generalizing Eqs. (2.8) and (2.9). This definition allows for an updated integration formula [33, 34]
Z
dV V⊗k⊗V∗⊗k = X
ν,µ∈Sk
Wg(q, ν−1µ)|νihµ|, (2.15) where Wg(q, ν) is the Weingarten function [33–35], and updated weight for- mulas
weighthti(σ, τ) = Wg(q2, τt−1σt) (2.16) weighthuvi(σ, τ) = qC(σu−1τv), (2.17) with C(ν) the function that returns the number of cycles in the permutation ν ∈ Sk.
Getting rid of negative weights by decimating incoming particles We now consider decimating the incoming particles; that is, in the partition function in Eq. (2.13), we explicitly perform the sum over τ. This removes the incoming particles from the system, giving a new stat mech system with half as many particles. The remaining sum over σ can still be interpreted as a partition function on this system, but the interactions become three-body instead of two-body: we now have an interaction hypergraph consisting of hyperedges on sets of three particles. These hyperedges huvwi exist whenever unitary u acts on a pair of qudits that were most recently acted upon by unitaries v and w. We can compute this hyperedge weight by summing over the k! possible values of τu, as follows:
weighthuvwi(σ) = X
τu∈Sk
qC(σ−1v τu)qC(σ−1w τu)Wg(q2, τu−1σu), (2.18) which for k = 2, yields the simple expression
weighthuvwi(σ) =
1 if σu =σv =σw
1
q+q−1 if σv 6=σw 0 if σu 6=σv =σw.
(2.19)
Figure 2.3: Decimation of incoming (blue) particles creates a three-body in- teraction between leftover outgoing particles.
The partition function for this version of the method reads EU[f(U)] =X
σ
Y
huvwi
weighthuvwi(σ). (2.20) At the expense of going from two-local to three-local interactions, we have arrived at a stat mech system with only non-negative interaction weights, which makes analysis easier. We can also see that this model is similar in spirit to the ferromagnetic Ising model since the largest weights occur when all of the spins agree.
Unfortunately, for k ≥ 3, some of the three-body weights can still be negative [36,37]. This observation, combined with the larger number of inter- nal states and more complex interaction weights makes analyzing systems for k ≥3much more difficult than for k = 2.
An equivalent picture of evolving n-bit configurations
In Chapter 4 and Chapter 5, we will essentially be analyzing the system de- scribed above with three-body weights, but we organize our analysis in a slightly different way. For any choice of t with 0≤t ≤s and each 0≤a < n, let ua,t be the maximum integer such that ua,t ≤ t and qudit a was one of the qudits involved in the gate at time step ua,t. Then we can construct what we call a configuration ~γ(t) ∈ {I, S}n by letting γa(t) =σua,t. The sequence of configurations γ = (~γ(0), ~γ(1), . . . , ~γ(s)) is called a trajectory, where~γ(t) is the same as~γ(t−1) except potentially at the positions that are involved in the tth gate. Each trajectory γ is associated with a microstate of the system in the three-body stat mech picture as long as γa(t)=γb(t) whenever unitaryt acts on qudits a and b. When this rule is obeyed, we can see from Eq. (2.19) that the overall weight of a trajectoryγ in the partition function decreases by a factor of q+q−1 each time gate t acts on two qudits that have different assignments in configuration~γ(t−1). This version of the stat mech method is derived in a self-contained manner in Chapter 4.