# Quantum Circuits II

## Some special kinds of circuits

austen.uk/slides/quantum-circuits-2-icts for slides

## Outline

• Circuits with special structure $\longrightarrow$ theoretical progress / new insights

• Random circuits

• Dual unitary circuits

• Other possibilities: Clifford (Arijeet’s talk), free fermions, …

$$Z_n(t)= \sum_{\mu_{1:N}=\{1,x,y,z\}^N} \mathcal{C}_{\mu_{1:N}}(t) \sigma_1^{\mu_1}\otimes\cdots \sigma_N^{\mu_N}$$

• Operator norm $\tr\left[Z_n^2(t)\right]=2$ is conserved under time evolution

$$\sum_{\mu_{1:N}=\{1,x,y,z\}^N} \mathcal{C}^2_{\mu_{1:N}}(t) = \frac{1}{2^{N-1}}$$

• Correlation function $\langle Z_j(t)Z_k(0)\rangle$ captures only a single coefficient

$$\langle Z_j(t)Z_k(0)\rangle=C_{jk}(t) \equiv \mathcal{C}_{1\cdots \mu_k=z \cdots 1}(t)$$

## Out of time order correlator

$$\operatorname{OTOC}_{jk}(t) \equiv \langle Z_j(t)Z_k(0)Z_j(t)Z_k(0)\rangle$$

• OTOC sometimes written as squared commutator

$$\langle \left[Z_k(0),Z_j(t)\right]^2 \rangle$$

• Relation between two expressions is (using $Z^2=1$)

$$\operatorname{OTOC}_{jk}(t) = \frac{1}{2}\langle \left[Z_k(0),Z_j(t)\right]^2 \rangle + 1$$

$$\operatorname{OTOC}_{jk}(t) = \frac{1}{2}\langle \left[Z_k(0),Z_j(t)\right]^2 \rangle + 1$$

• At short times commutator vanishes so $\operatorname{OTOC}_{jk}(t\to 0)=1$

$$\operatorname{OTOC}_{jk}(t)= 2^{N-1} \sum_{\mu_{1:N}}\mathcal{C}_{\mu_{1:N}}^2(t)\left[\delta_{\mu_k,0}+\delta_{\mu_k,3}-\delta_{\mu_k,1}-\delta_{\mu_k,2}\right]$$

• $\operatorname{OTOC}_{jk}(t)\neq 1$ after operator $Z_j(t)$ spreads from site $j$ to site $k$

• Characteristic speed of propagation of OTOC is “butterfly velocity” $v_\text{B}$

• Since OTOC depends on square of the coefficients, a nonzero value survives after averaging over random circuits

• OTOC measured in 2021 by Google

## Remark: operator entanglement

• OTOC provides one measure of operator spreading

• Another question: how many nonzero coefficients $\mathcal{C}_{\mu_{1:N}}$?

• Introduce Schmidt decomposition for operators $$\mathcal{O}_{AB} = \sum_{n=1}^{\min(n^2_A, n^2_B)} \Sigma_n A_n\otimes B_n$$

• $\Sigma_n\geq 0$ are operator Schmidt coefficients

• $A_n$ and $B_n$ are orthonormal operators on $\mathcal{H}_A$ and $\mathcal{H}_B$ i.e. $\tr\left[A^\dagger_m A_n\right]=\tr\left[B^\dagger_m B_n\right]=\delta_{mn}$

• Same entanglement measures as before be applied to evaluate the operator entanglement

• Simplest example, analogous to Bell state, is SWAP operator

$$\operatorname{\mathsf{SWAP}}=\frac{1}{2}\left[X\otimes X+Y\otimes Y+Z\otimes Z + \mathsf{1}\otimes\mathsf{1}\right]$$

• All Schmidt coefficients are equal (maximum operator entanglement)

## Random circuits

• Met this idea last time: average over $\theta$ in

$$U_{j,j+1} = \cos\theta \mathbb{1}_{j,j+1} + i\sin\theta \operatorname{\mathsf{S}}_{j.j+1}$$

• Now consider even more random gates: average uniformly over single site unitaries

## Why?

1. No symmetries so results should be generic (“in some sense”)

2. (real reason) Averaging over circuits simplifies things considerably, sometimes allowing tractable classical description

• Choose gates iid. Other options: randomness in space but not time

• Recent review: Fisher et al (2023)

### OTOC in random circuits

$$\operatorname{OTOC}_{jk}(t) \equiv \langle Z_j(t)Z_k(0)Z_j(t)Z_k(0)\rangle$$

• OTOC can be extracted by appropriately contracting indices in $Z_j(t)\otimes Z_j(t)$ (two copies)

• When we average over random unitaries, only certain components survive

• Single qubit unitaries identified with rotations, so look for scalar operators made from two copies on each site

$$\mathsf{1}\otimes\mathsf{1},\qquad \frac{1}{3}\left[X\otimes X+Y\otimes Y + Z\otimes Z\right]$$

• Different papers prefer different bases. The most popular choice is $$\mathsf{1}\otimes\mathsf{1},\qquad \operatorname{SWAP}$$ (recall that $\operatorname{\mathsf{SWAP}}=\frac{1}{2}\left[X\otimes X+Y\otimes Y+Z\otimes Z + \mathsf{1}\otimes\mathsf{1}\right]$)

• Advantage: generalizes to multiple copies

• General set of invariants are generalized SWAP operators corresponding to permutations of copies (for two copies only two permutations)

• Now use invariant local basis to expand average of two copies

$$\mathcal{O}^{(2)}(t)=\overline{O(t) \otimes O(t)} \equiv \overline{O(t)^{\otimes 2}}$$

\begin{align*} \mathsf{O} &\equiv\mathsf{1}\otimes\mathsf{1} \\ \mathsf{1}&\equiv\frac{1}{3}\left[X\otimes X + Y\otimes Y+ Z\otimes Z\right] \end{align*}

• Introduce basis $\mathsf{S}_{1:N}\equiv\mathsf{S}_1\otimes \mathsf{S}_2\otimes\cdots \mathsf{S}_N$, with $\mathsf{S}_j=\mathsf{0},\mathsf{1}$

$$\mathcal{O}^{(2)}(t) = \sum_{\mathsf{S}_{1:N}\in\{\mathsf{0},\mathsf{1}\}^N} P_{\mathsf{S}_{1:N}}(t)\mathsf{S}_{1:N}$$

• Coefficients $P_{\mathsf{S}_{1:N}}(t)$ describe averaged OTOC
• Next find how $P_{\mathsf{S}_{1:N}}(t)$ are updated by a single gate (after averaging)

• Gate acting on sites $j$ and $j+1$ yields

$$U^\dagger_{j,j+1}\otimes U^\dagger_{j,j+1} \mathcal{O}^{(2)}(t)U_{j,j+1}\otimes U_{j,j+1}$$

• Take $U_{j,j+1}$ of form $$U_{j,j+1} = V_{j,j+1} u_j \otimes u_{j+1}$$ where $u_j$ and $u_{j+1}$ are single quibit unitaries chosen uniformly

• After averaging all non-invariant components vanish and invariant components don’t depend on $u_j$ and $u_{j+1}$

• Extract $P_{\mathsf{S}_{1:N}}(t+1)$ using orthgonality $\tr\left[\mathsf{O}\mathsf{1}\right]=0$

$$P_{\mathsf{S}_{1:N}}(t+1) = \sum_{\mathsf{S}’_j, \mathsf{S}’_{j+1}} P_{\mathsf{S}_1\cdots \mathsf{S}’_j \mathsf{S}’_{j+1}\cdots \mathsf{S}_N}(t)\Omega_{\mathsf{S}’_j \mathsf{S}’_{j+1},\mathsf{S}_j \mathsf{S}_k}$$

• Precise form of matrix $\Omega$ depends on $V_{j,j+1}$ “core”

• Use conservation of operator norm $\tr\left[O(t)^\dagger O(t)\right]$

$$\overline{\tr\left[O(t)^\dagger O(t)\right]} = 2\sum_{\mathsf{S}_{1:N}\in{\mathsf{0},\mathsf{1}}^N} P_{\mathsf{S}_{1:N}}$$

$$\sum_{S_j, S_{j+1}}\Omega_{\mathsf{S}’_j \mathsf{S}’_{j+1},\mathsf{S}_j \mathsf{S}_k} = 1$$

• If matrix elements additionally nonnegative we have a Markov process, with transition matrix $\Omega$

\begin{align*} \Omega&=\left(\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1-a-b & a & b \\ 0 & a & 1-a-b & b \\ 0 & \frac{b}{3} & \frac{b}{3} & \left(1-\frac{2}{3} b\right) \end{array}\right) \\ a&=\frac{1}{3}\left(2 \sin ^{2} \theta+\sin ^{4} \theta\right) \qquad b=\frac{1}{3}\left(\frac{1}{2} \sin ^{2} 2 \theta+2\left(\sin ^{2} \theta+\cos ^{2} \theta\right)\right) \end{align*}

• $\theta=\pi/2$ for $i\operatorname{SWAP}$ gate and $\theta=\pi/4$ for $\sqrt{i\operatorname{SWAP}}$

## Remarks

• Idea that unitary averages over two copies yields a Markov process on the invariant space goes back to Oliveria, Dahlsten, and Plenio (2007), which was concerned with dynamics of average purity $\gamma\equiv \tr \rho_A^2$

• $\bar \gamma$ can be extracted from average of two copies of density matrix $\rho(t)\otimes\rho(t)$. See e.g. Rowlands and Lamacraft (2018) for noisy unitary evolution in continuous time

## The Markov process

\begin{align*} \Omega&=\left(\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1-a-b & a & b \\ 0 & a & 1-a-b & b \\ 0 & \frac{b}{3} & \frac{b}{3} & \left(1-\frac{2}{3} b\right) \end{array}\right) \end{align*}

• Describes transitions

$$\mathsf{10} \xleftrightharpoons[a]{a} \mathsf{01} \qquad \mathsf{11} \xleftrightharpoons[b/3]{b} \mathsf{10},\mathsf{01}$$

• Note that $\mathsf{0}=\mathsf{1}\otimes\mathsf{1}$ is “inert”: there no transitions from or to $\mathsf{00}$

## Fredrickson–Andersen model

$$\mathsf{10} \xleftrightharpoons[a]{a} \mathsf{01} \qquad \mathsf{11} \xleftrightharpoons[b/3]{b} \mathsf{10},\mathsf{01}$$

• Stationary state: independent sites with $p_1=3/4$, $p_0=1/4$

## Butterfly velocity

• Front propagation characterised by finite velocity $v_\text{B}$

• Front broadens unless $v_\text{B}$ maximal as for $i\operatorname{SWAP}$
• Diffusive in 1D $\propto \sqrt{t}$

• KPZ dynamics in 2D

## Classical simulation?

• Efficient simulation of averaged OTOC dynamics via Monte Carlo

• Appearance of Markov process a little surprising

## OTOC fluctuations

• Circuit-to-circuit fluctuations of OTOC from

$$\mathcal{O}^{(4)}(t)=\overline{O(t) \otimes O(t) \otimes O(t) \otimes O(t)} \equiv \overline{O(t)^{\otimes 4}}$$

• Go through same procedure of identifying invariant states

• Evolution of average now involves negative matrix elements

• Leads to sign problem in Monte Carlo simulation

• Same problem for $\overline{\operatorname{OTOC}}$ in models with number conservation (Rowlands and Lamacraft)

## Recall: kicked Ising model

• Time dependent Hamiltonian with kicks at $t=0,1,2,\ldots$.

\begin{align*} H_{\text{KIM}}(t) = H_\text{I}[\mathbf{h}] + \sum_{n}\delta(t-n)H_\text{K}\\ H_\text{I}[\mathbf{h}]=\sum_{j=1}^L\left[J Z_j Z_{j+1} + h_j Z_j\right],\qquad H_\text{K} &= b\sum_{j=1}^L X_j \end{align*}

## Entanglement Growth for KIM

$$\lim_{L\to\infty} S_A =\min(2t-2,N_A)\log 2,$$

• Any $h_j$; initial $Z_j$ product state

## Entanglement Spectrum

• Rényi entropies depend on eigenvalues of reduced density matrix

$$S^{(\alpha)}_A = \frac{1}{1-\alpha}\log \text{tr}\left[\rho^\alpha\right]=\frac{1}{1-\alpha}\sum_n p_n^\alpha$$

• For SDKIM have $2^{\min(2t-2,N_A)}$ non-zero eigenvalues all equal

$$p_n = \left(\frac{1}{2}\right)^{\min(2t-2,N_A)}$$

## Thermalization

• After $N_A/2 + 1$ steps, reduced density matrix is $\propto \mathbb{1}$

• All expectations (with $A$) take on infinite temperature value

## Dual unitarity

• Recall KIM has circuit representation

\begin{aligned} \mathcal{K} &= \exp\left[-i b X\right]\\ \mathcal{I} &= \exp\left[-iJ Z_1 Z_2 -i \left(h_1 Z_1 + h_2 Z_2\right)/2\right] \end{aligned}

• At $|J|=|b|=\pi/4$ has additional property of dual unitarity

• Unitarity:

## $\rho_A$ via dual unitarity

• Initial state of NN Bell pairs

• 8 sites; 4 layers

• $\rho_A$ is unitary transformation of

$$\mathbb{1}\otimes\mathbb{1}\otimes\mathbb{1}\otimes\mathbb{1}\otimes\mathbb{1}\otimes\mathbb{1}\otimes\mathbb{1}\otimes\mathbb{1}$$

## Shallower…

• $\rho_A$ is unitary transformation of

$$\mathbb{1}\otimes\mathbb{1}\ket{\Phi^+}\bra{\Phi^+}\otimes\ket{\Phi^+}\bra{\Phi^+}\otimes\mathbb{1}\otimes\mathbb{1}$$

## General case

• RDM is unitary transformation of

$$\rho_0=\overbrace{\frac{\mathbb{1}}{2}\otimes \frac{\mathbb{1}}{2} \cdots }^{t-1} \otimes\overbrace{\ket{\Phi^+}\bra{\Phi^+} \cdots }^{N_A/2-t+1 } \otimes \overbrace{\frac{\mathbb{1}}{2}\otimes \frac{\mathbb{1}}{2} \cdots }^{t-1}$$

• RDM has $2^{\min(2t-2,N_A)}$ non-zero eigenvalues all equal to $\left(\frac{1}{2}\right)^{\min(2t-2,N_A)}$

• Converse – maximal entanglement growth implies dual unitary gates – recently proved by Zhou and Harrow (2022)

## The dual unitary family

• $4\times 4$ unitaries are 16-dimensional

• Family of dual unitaries is 14-dimensional

• Includes kicked Ising model at particular values of couplings

• Dual unitaries not “integrable” but have enough structure to allow many calculations

## ‘KIM’ property

• ($q=2$ here) Not satisfied by e.g. $\operatorname{SWAP}$

• Maps product states to maximally entangled (Bell) states

• Product initial states also work for KIM!

• Piroli et al (2020) studied more general initial states

• Foligno and Bertini (2023) study general initial conditions

## Correlation functions

• Infinite temperature correlator $\tr\left[\sigma^\alpha_x(x,t)\sigma^\beta(y,0)\right]$