$$ \nonumber \newcommand{\br}{\mathbf{r}} \newcommand{\bR}{\mathbf{R}} \newcommand{\bp}{\mathbf{p}} \newcommand{\bk}{\mathbf{k}} \newcommand{\bq}{\mathbf{q}} \newcommand{\bv}{\mathbf{v}} \newcommand{\bx}{\mathbf{x}} \newcommand{\bz}{\mathbf{z}} \DeclareMathOperator*{\E}{\mathbb{E}} $$

Quantum Circuits of Dual Unitary Gates

Sarang Gopalakrishnan and AL 2019

Pieter Claeys and AL 2020, and arXiv:2009.03791

Slides at austen.uk/slides/dual-unitaries-pcts


A circuit


Unitarity and dual unitarity

A circuit A circuit

  • For qubits dual unitary gate has 14 parameters (c.f. 16 for $U(4)$!)


  • Origins of dual unitarity: entanglement
  • Correlation functions
  • OTOCs

Origins of dual unitarity

Kicked Ising Model

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

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

  • “Stroboscopic” form of $U(t)=\mathcal{T}\exp\left[-i\int^t H_{\text{KIM}}(t’) dt’\right]$

$$ \begin{aligned} U(n_+) = \left[U(1_+)\right]^n,\qquad U(1_-) = K I_\mathbf{h}\\ I_\mathbf{h} = e^{-iH_\text{I}[\mathbf{h}]}, \qquad K &= e^{-iH_\text{K}}, \end{aligned} $$

Entanglement Growth for Self-Dual KIM

$$ \lim_{L\to\infty} S^{(n)}_A(t) =\min(2t-2,N)\log 2, $$

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

Entanglement Spectrum

  • Rényi entropies depend on eigenvalues of reduced density matrix

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

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

$$ \lambda_\alpha = \left(\frac{1}{2}\right)^{\min(2t-2,N)} $$

KIM as a circuit

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

Properties of KIM gate

  • When $|J|=|b|=\pi/4$ KIM gate is dual unitary

  • This allows for simple proof of entanglement dynamics!

Gopalakrishnan and Lamacraft (2019)

Graphical representation of density matrix

  • $\rho(t) = U\rho_0 U^\dagger$, working from middle out

Folded picture

  • Reduced density matrix: trace over all but $N$ sites (here $N=6$)

Using unitarity…

Go further with dual unitarity?

Initial conditions

  • For SDKIM we have in addition

  • Just the thing for $\rho_0 = \otimes_j |Z_j\rangle\langle Z_j|$!

…Using dual unitary gates

  • A unitary applied to

$$ \overbrace{\frac{\mathbb{1}}{2}\otimes \frac{\mathbb{1}}{2} \cdots }^{t-1} \otimes\overbrace{|Z_1\rangle\langle Z_1|\otimes |Z_2\rangle\langle Z_2| \cdots }^{N-2t+2 } \otimes \overbrace{\frac{\mathbb{1}}{2}\otimes \frac{\mathbb{1}}{2} \cdots }^{t-1} $$

Result for RDM

  • Spectrum of RDM same as spectrum of

$$ \overbrace{\frac{\mathbb{1}}{2}\otimes \frac{\mathbb{1}}{2} \cdots }^{t-1} \otimes\overbrace{|Z_1\rangle\langle Z_1|\otimes |Z_2\rangle\langle Z_2| \cdots }^{N-2t-2 } \otimes \overbrace{\frac{\mathbb{1}}{2}\otimes \frac{\mathbb{1}}{2} \cdots }^{t-1} $$

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

More general initial conditions

  • Piroli, et al., (2020) considered 2-site MPS initial conditions that allow for solution in similar way

  • For $2t>N $ result is the same: $\infty$-temperature RDM

  • At earlier times structure of initial state important

Correlation functions ($T=\infty$)

$$ c(x,y,t)=\mathop{\text{tr}}\left[U(t)^\dagger O(x)U(t) O(y)\right] $$

Chan, De Luca, Chalker (2018)

Using unitarity

“Folded” picture

  • Later operator must be in “future light cone” of earlier

On the light cone

Light cone quantum channel

$$ \begin{align} c_\nu^{\alpha\beta}(\nu t,t) = \frac{1}{q} {\rm tr}\left[\mathcal M_{\nu}^{2t}(a^\beta)a^\alpha\right]\\ \mathcal M_{+}(a) = \frac{1}{q} {\rm tr}_1\left[U^\dagger (a\otimes\mathbb{1}) U\right] \end{align} $$

  • Unitarity means map is CPTP and unital (identity fixed)
  • See Bertini, Kos, Prosen (2019) for dual unitary case, but construction is general

‘Typical’ correlations

  • Generic: decay governed by dominant eigenvalue of channel
  • Can evaluate correlations inside light cone, just need a bigger channel (see later)!

Dual unitary case

  • Unitarity implies future and past lightcones up and down
  • Dual unitarity implies future and past left and right
  • Conclusion: correlations only on light cone

Bertini, Kos, Prosen (2019)


$$ C_{\alpha \beta}(x,t) = \langle \sigma_{\alpha}(0,t) \sigma_{\beta}(x,0) \sigma_{\alpha}(0,t) \sigma_{\beta}(x,0) \rangle. $$

  • $C_{\alpha \beta}(|x|>t,t)=1$ since $\left[\sigma_{\alpha}(0,t),\sigma_{\beta}(x,0)\right]=0$ for $|x|>t$

  • For smaller $|x|$ OTOC deviates from 1.

  • Limiting value of $|x|/t$ where this occurs defines butterfly velocity

  • Example: random unitary circuits with local dimension $q$ have $$ v_\text{B} = \frac{q^2-1}{q^2+1} $$ $v_\text{B}\to 1$ as $q\to\infty$

Maximum velocity quantum circuits

  • Which circuits have the largest butterfly velocity $v_\text{B}=1$?
  • Natural guess: dual-unitary circuits!


(and no broadening)


Fold it up…

  • Light cone coordinates $n_+ = (t+ x)/2$, $n_- = (t- x+2)/2$

$$ \begin{align} C^{+}_{\alpha \beta }(x,t) = \left(L(\sigma_{\alpha})\right|\left(T_{n_-}\right)^{n_+}\left|R^{-}(\sigma_{\beta})\right), \nonumber\\ \end{align} $$

  • This is for $x-t$ even, similar expression for $x-t$ odd

Transfer matrix

  • Using only unitarity

    • Find eigenstate of transfer matrix with eigenvalue one
    • Show that $\lim_{m\to \infty} \left(L_n(\sigma_{\alpha})\right|\left(T_n\right)^m\left|R^{\pm}(\sigma_{\beta})\right) = 1$
  • This is the value outside light cone, so $v_\text{B}<1$ generically

  • $v_\text{B}=1$ requires additional eigenstates with eigenvalue one

Using dual unitarity…

  • Find $n+1$ eigenvectors with eigenvalue 1
  • For even $n$, $C^{+}_{\alpha \beta}(x,t)$ vanishes inside light cone $$ \begin{align} \lim_{(x+t) \to \infty }C^{+}_{\alpha \beta}(x,t) = \begin{cases} -\frac{1}{q^2-1} \qquad &\text{if} \quad x=t,\\ 0 \qquad &\text{if} \quad x \neq t. \end{cases} \end{align} $$

  • For odd $n$, $C^{-}_{\alpha \beta}(x,t)$ can be found in terms of channel $\mathcal M_{\pm}(\cdot)$ describing light cone correlator

  • Range of behaviour moving inside light cone

Further special cases…

  • Integrable SDKIM: Exponentially many eigenvectors with eigenvalue one. OTOC immediately saturates to constant value inside light cone.

  • Are all maximal velocity circuits dual unitary? No! Found a kicked XY model with $v_\text{B}=1$


  • Dual unitary circuits are a big solvable family with a diverse phenomenology that includes integrable and chaotic behaviour…

  • … but maybe not that diverse for such a big family! $v_\text{B}=1$ always, for example

  • Still more to do: effect of measurements, coding, computational complexity…