Next Generation Reservoir Computing
quantum systems
By
For
Ninnat Dangniam
2024 KMUTT-IF Winter School
19 Dec 2024
Quantum Machine Intelligence, 6, 57 (2024)
& manuscript in preparation
NG-RC
Krai Cheamsawat
Apimuk Sornsaeng
Thiparat Chotibut
Paramott Bunnjaweht
Now at
Now at
the (local) team
and more...
A new way to predict properties of quantum states
Heuristic, training-based
Type of algorithm
Type of data
quantum state tomography
- Quantum states of \(N\) qubits are complex vectors in \(2^N\) dimensions—an exponential large object
- Measuring the state gives one of the \(2^N\) outcomes The state is destroyed after the measurement
- Therefore, learning quantum states always required multiple identical copies—\(\Theta(4^N)\), in fact
\ket{\psi} = \sum_{\vec{x}} \alpha_{\vec x} \ket{\vec{x}=x_1 x_2 \dots x_N}
\(x_j=\) 0 or 1
\alpha_{\vec{x}} \in \mathbb{C}
\ket{\psi}_{n}
\ket{\psi}_{n-1}
\dots
\ket{\psi}_1
\psi
Classical description of \(\ket{\psi}\)
Reconstruction algorithm
quantum state tomography
- Learning quantum states also require several independent observables
- This is best illustrated in the qubit case
\sigma^x = \begin{pmatrix}0&1\\1&0 \end{pmatrix}
\sigma^y = \begin{pmatrix}0&\!\!\!-i\\i&0 \end{pmatrix}
\sigma^z = \begin{pmatrix}1&0\\0&\!\!\!-1 \end{pmatrix}
\ket{\psi} = \alpha \ket{0} + \beta\ket{1}
|\alpha|^2 + |\beta|^2=1
\langle O \rangle \equiv \bra{\psi}O\ket{\psi}
\begin{align*}
\langle \sigma^x \rangle &= 2\textrm{Re}(\alpha\beta^*) \\
\langle \sigma^y \rangle &= 2\textrm{Im}(\alpha\beta^*) \\
\langle \sigma^x \rangle &= |\alpha|^2 - |\beta|^2
\end{align*}
\ket{0}
\ket{1}
\begin{bmatrix}
\langle \sigma^x \rangle \\
\langle \sigma^y \rangle \\
\langle \sigma^z \rangle
\end{bmatrix}
=
\begin{bmatrix}
\sin\theta\cos\varphi \\
\sin\theta\sin\varphi \\
\cos\theta
\end{bmatrix}
quantum state tomography
quantum state tomography
quantum state tomography
quantum state tomography
quantum state tomography
quantum state tomography
- Quantum tomography reconstructs a quantum state from its properties or "shadows" \(\langle O_j \rangle\)
- But if the state is a tool for calculating shadows, why don't we estimate shadows directly from shadows?
Allegory of Plato's cave
- This shadow tomography task (Aaronson 2018, coined by Steve Flammia) turns out to be much more manageable
Exp
polynomial
4edges, Wikimedia
shadow tomography
Prediction algorithm
\{\langle O_j \rangle \}
\{ \langle \widetilde{O}_j \rangle \}
deterministic chaos
Wikimol, Wikimedia
\begin{align*}
\dot{x} &= \sigma(y-x) \\
\dot{y} &= x(\rho-z)-y \\
\dot{z} &= xy-\beta z
\end{align*}
Model of atmospheric convection governed by nonlinear DEs
Exhibits hypersensitivity to initial conditions (the "butterfly effect")
Lorenz attractor
\rho=28, \sigma=10, \beta=\frac{8}{3}
it may happen that small differences in the initial conditions produce very great ones in the final phenomena [...] Prediction becomes impossible, and we have the fortuitous phenomenon.
Henri Poincaré
deterministic chaos
Lyapunov time is a characteristic timescale at which nearby trajectories diverge
next-generation reservoir computing
NG-RC accurately predicts the Lorenz attractor up to ~5 Lyapunov times (1.1 unit time in the figure). How?
next-generation reservoir computing
Nonlinear transformations
Stirring liquid surface = computation
next-generation reservoir computing
o_k = s_k \oplus s_{k-{\color{yellow}\Delta}} \oplus s_{k-2 {\color{yellow}\Delta}} \oplus \cdots \oplus s_{k-({\color{pink}m}-1){\color{yellow}\Delta}}
- NG-RC injects generic (polynomial) nonlinearity by constructing \(m\)-time-delay vectors out of observed time series \(\{s_0, s_1, \dots, s_{T-1}\}\)
s_k = \begin{bmatrix}
a_1 \\ \vdots \\ a_M
\end{bmatrix}
\underbrace{x_k = o_k \oplus (o_k)^{\otimes {\color{lightgreen}p}}}
a_j \in \mathbb{R}
3 hyperparameters
Step size \(\Delta\)
Time delay \(m\)
Nonlinearity degree \(p\)
dim \(mM+(mM)^p\)
- \(x_k\) is then "trained" by a linear transformation that minimizes a (regularized) least-square loss w.r.t. the target
vs
NG-RC gives a heuristic for "time-translated shadow tomography"*
Prediction algorithm
\{\langle O_j \rangle \}
\{ \langle \widetilde{O}_j \rangle \}
Future
Past
*We don't claim to solve the shadow tomography task in its rigorous formulation
What is necessary for accurate prediction?
- Time delay \(m\) and the number of observables \(M\) should obey the Takens' condition (Sauer et al. 1991) related to attractors in late-time dynamics
- Nonlinearity degree \(p\) should be related to the nonlinearity of DEs that govern the observables' dynamics (Zhang et al. 2023)
Some key messages:
s_k = \begin{bmatrix}
\langle O_1\rangle \\ \vdots \\ \langle O_M\rangle \end{bmatrix}
\langle O_j \rangle \in \mathbb{R}
Ng-rc for quantum
Jpagett, Wikimedia
Bose-Hubbard model
1D spin chains
Heisenberg XXZ model
Tilted-field Ising model (TIM)
AG-FKT, TU Braunschweig
H = -J \sum_{i=0}^{N-1} (\sigma_i^x \sigma_{i+1}^x + \sigma_i^y \sigma_{i+1}^y + g\sigma_i^z \sigma_{i+1}^z)
H = -J \sum_{i=0}^{N-1} [ \sigma_i^z \sigma_{i+1}^z
+ g(\sigma_i^x \sin\theta + \sigma_i^x \cos\theta)]
H = -J \sum_{i,j=0}^{N-1} J a^{\dagger}_i a_i + \sum_{i=0}^{N-1} \left(
\epsilon a^{\dagger}_i a_i + \frac{U}{2}a^{\dagger2}_i a_i^2 +F(a^{\dagger}_i + a_i) \right)
Quantum chaotic
Quantum integrable
Linear EOMs
\frac{d}{dt}\langle \sigma_n \rangle = f_{\textrm{linear}} \left( \left\{ \langle \sigma_{n-1} \rangle ,\langle \sigma_{n} \rangle ,\langle \sigma_{n+1} \rangle \right\}
\right)
\frac{d\alpha_j}{dt} = -(\gamma+i\Delta) \alpha_j - i\sum_{k=0}^{N-1} J_{jk} \alpha_k - iU\alpha_j |\alpha_j|^2 - iF
Non-linear EOMs (semiclassical limit)
model systems
takens' embedding theorem
\(\Omega\) = Tomographically complete set of observables
\(S_M \subset \Omega\) = Accessible observables
\mathbb R^{|\Omega|}
\mathcal A
\mathbb R^{M}
Attractor
\mathbb R^{mM}
Generic \(F\) = one-to-one embedding of the attractor provided
\(mM > 2\dim\mathcal (A)\)
Trade time dimension for number of observables!
F
takens' embedding theorem
XXZ model
Tilted Ising model
(NRMS) Error
m\ge69
- Thus, for a fixed number of observables \(M\), there is a minimal embedding dimension \(m_{\textrm{Takens}} > 2\dim\mathcal (A)/M \)
- For the spin chain models, the EOMs of accessible observables in \(S_M\) (one-site & nearest neighbor two-site Pauli observables) is not closed, nevertheless \(\max \dim \mathcal(A)\) is at most the number of (real) parameters of (pure) quantum states: \(2\cdot 2^N-2\)
\frac{d}{dt}\langle \sigma_n \rangle = f_{\textrm{linear}} \left( \left\{ \langle \sigma_{n-1} \rangle ,\langle \sigma_{n} \rangle ,\langle \sigma_{n+1} \rangle \right\}
\right)
N=8, M=15, p=1
takens' embedding theorem
|\Omega|=20
M_{\textrm{one-site}} = 2 \implies m_{\textrm{Taken}}>20
M_{\textrm{two-site}} = 4 \implies m_{\textrm{Taken}}>10
In the semiclassical limit, observables are c-numbers \(\alpha_j = x_j + ip_j\)
p>1
Nonlinear EOMs
NG-RC is a heuristic, training-based "time-translated shadow tomography"
Prediction algorithm
\{\langle O_j \rangle \}
\{ \langle \widetilde{O}_j \rangle \}
Future
Past
Prior knowledge about the dynamics
optimization
!
- We demonstrates a capability of NG-RC to predict quantum properties in quantum many-body systems
- Performance can be understood via Takens' embedding theorem
- However, hyperparameter optimization would require prior knowledge about the dynamics we want to predict, a catch-22 situation! (Zhang et al. 2023)
- There're a lot of questions to think about e.g.
- What do attractors of quantum observables look like?
- How to choose informative observables?
conclusion
Quantum NG-rc
- NG-RC algorithm can be quantized and works in the same way (but coherently takes quantum input and output instead)
- Block encoding saves the space for storing \(\underbrace{x_k = o_k \oplus (o_k)^{\otimes {\color{lightgreen}p}}} \) exponentially
- Iterative prediction is costly, so we invented skip-ahead method
- However, inputting quantum data can be inefficient
dim \(mM+(mM)^p\)
Please see the paper if you're interested!
skippa skippa
Thank you
fractal dimensions
Generalized Takens' theorem in Sauer et al. allow fractal attractors, in which case the relevant dimension is the box-counting dimension
\dim_B = \lim_{\epsilon\to0} \frac{\log \mathcal N(\epsilon)}{\log \epsilon^{-1}}
\( \mathcal N(\epsilon) = \) the number of boxes of size \( \epsilon \) required to cover the attractor
\mathcal N(\epsilon) \sim \epsilon^{-\dim_B}
For dynamical systems, \(\dim_B\) is estimated by generating a trail of points in the attractor and count the number of boxes they visit, then plotting a log-log plot of \(\mathcal N(\epsilon)\) vs \( \epsilon \)
The estimate is sensitive to statistical noise at small \(\epsilon\)
fractal dimensions
\dim_C = \lim_{\epsilon\to0} \frac{\log \mathcal C(\epsilon)}{\log \epsilon}
\mathcal N(\epsilon) \sim V \epsilon^{-d_B}
C(\epsilon) = \frac{2}{N(N-1)} \sum_{i< j}^N \Theta(\epsilon-|x_i-x_j|)
C(\epsilon) \sim \epsilon^{\dim_C}
More robust to the statistical noise is the correlation dimension
\dim_C \le \dim_B
Tilted Ising
Bose-Hubbard
Quantum NG-rc
U_A = \quad\quad\begin{bmatrix} A/\alpha & & \cdot & \\ \cdot & & \cdot & \end{bmatrix}
\begin{matrix} |0\rangle^{\otimes a} \\ |0\rangle^{\otimes a}_{\perp} \end{matrix}
\begin{matrix} |0\rangle^{\otimes a} & |0\rangle^{\otimes a}_{\perp} \end{matrix}
\lVert A - \langle 0|^{\otimes a}U|0\rangle^{\otimes a} \rVert \le \epsilon
\(U_A\) is said to be a block-encoding of \(A\)
\( (\alpha,a,\epsilon) \)-approximate
NG-RC for quantum systems, by quantum systems (2024 KMUTT-IF)
By Ninnat Dangniam
NG-RC for quantum systems, by quantum systems (2024 KMUTT-IF)
2024 Winter School KMUTT
