State reachability in isolated systems with 2-body Hamiltonians

Konrad Szymański

Universität Siegen

This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 945422.

All states are reachable with time-dependent 2-body unitaries…

https://texample.net/tikz/examples/quantum-circuit/
\[U=\mathcal{T}\exp(-i{\color{#d70}\int H(t)}\operatorname{d}t),\] where \[\color{#d70}H(t)\color{black}=\sum_{i,j} A^{(i)}(t)\otimes B^{(j)}(t),\] if arbitrary local operators are permitted.

What about autonomous 2-body evolution?

Q1: Can every $\color{#00c}\ket\psi$ be arbitrarily well approximated by \[\color{#a00}\ket{\phi(t)}\color{black}=\exp(-i{\color{#d70}H}t)\underbrace{\bigotimes_{k}\color{#090}{\ket{\phi_k}}}_{\color{#090}\ket\phi},\] with \(\color{#d70}H\color{black}=\sum_{i,j} A^{(i)} \otimes B^{(i)}\)?

I.e. for every $\varepsilon>0$, does there exist $\color{#090}\ket\phi$ and $\color{#d70}H$ such that $\lvert \color{#00c}\ket\psi\color{black}-\exp(-i\color{#d70}H\color{black})\color{#090}\ket\phi\color{black}\rvert<\varepsilon$?

Parameter count: unreachable states for $N\ge 7$ qubits.

Any $N$ qubit quantum state: \[\color{#00c}\ket\psi\color{black} = \!\!\!\!\!\!\!\!\!\!\sum_{b_1,\ldots,b_N \in \{0,1\}}\!\!\!\!\!\!{\color{#00c}c_{\vec b}} \ket{b_1,\ldots,b_N}.\] $\color{#00c}2^{N}-1$ complex = $\color{#00c}2^{N+1}-2$ real parameters.
$2$-body evolution of a product state $\exp(-i \color{#d70}H\color{black})\bigotimes_k \color{#090}\ket{\phi_k}$: \[\color{#d70}H\color{black}=\sum_{i,j,\alpha,\beta} \color{#909}\gamma_{i,j,\alpha,\beta}\color{black} \sigma^{(i)}_\alpha\sigma^{(j)}_\beta + \sum_{i,\alpha} \color{#660}h_{i,\alpha}\color{black} \sigma^{(i)}_\alpha.\] $\color{#909}{N\choose 2} 3^2\color{black}+\color{#660}3N\color{black}+\color{#090}2N\color{black}=\color{#a00}\frac{N(9N-1)}2$ real parameters.

But: space-filling curves exist!

Almost all `interesting' states seem to be reachable

`Interesting': highly entangled, structured ones.

  • Graph states: explicitly \[\color{#00c}\ket{G}\color{black}=\exp\left(-i\pi \sum_{(i,j)\in E(G)} \color{#d70}\frac{\sigma_z^{(i)}+1}2 \frac{\sigma_z^{(j)}+1}2\color{black}\right) \color{#090}\ket{+}^{\otimes{N}}.\]
  • Dicke states: $\ket{D_{N,k}} \propto \ket{\underbrace{1,\ldots,1}_{k},\overbrace{0,\ldots,0}^{N-k}}+\text{permutations}$

Not all states are eigenstates of 2-body Hamiltonians.

Haselgrove (2003):

  • Let $\color{#00c}\ket\psi$ be the codespace of a $[[N,0,2]]$ quantum error correcting code. \[\color{#d70}\sigma^{(i)}_\alpha \sigma^{(j)}_\beta\color{#00c} \ket\psi\color{black}\perp\color{#00c} \ket\psi.\]
  • 2-body $\color{#d70}H$ has only Pauli strings of length $\le 2$.
  • Hence, $\color{#d70}H\color{#00c}\ket\psi\color{black}\perp\color{#00c}\ket\psi$, and $\color{#00c}\ket\psi$ is not an eigenstate of $\color{#d70}H$.
  • Distance from any eigenstate $\color{#770}\ket k$: \[\lvert\color{#00c}\ket{\psi}\color{black}-\color{#770}\ket{k}\color{black}\rvert\ge\left(\frac{9N^2-3N+2}{2}\right)^{-1/2}.\]

Not all unitaries are generated by 2-body dynamics.

An unitary $U=\sum_k e^{i\alpha_k} \color{#770}\ket{k}\bra{k}$ is reachable
$\implies$
$\{\color{#770}\ket{k}\color{black}\}$ are eigenstates of some 2-body $\color{#d70}H$.

What is reachable in small time?

Intuitively, only 2-body correlations are produced…

Sector lengths: \[A_1({\color{#00c}\ket\psi}) = \sum_{i,\alpha} \braket{\sigma^{(i)}_\alpha}_{{\color{#00c}\psi}}^2.\] \[A_2({\color{#00c}\ket\psi}) = \sum_{i,j,\alpha,\beta} \braket{\sigma^{(i)}_\alpha\sigma^{(j)}_\beta}_{{\color{#00c}\psi}}^2.\] \[A_k({\color{#00c}\ket\psi}) = \sum\braket{\text{length-}k\text{ Pauli string}}^2_{{\color{#00c}\psi}}.\]
All $N$-qubit pure product states $\color{#090}\bigotimes_k \ket{\phi_k}$ have \[A_k = \color{#090}{N \choose k}.\] What is $\color{#a00}A_k(\exp(-i\varepsilon H)\ket{\phi})$? \[\color{#a00}A_k\color{black} = {N\choose k} + \lambda \left(\binom{N-2}{k-1}-\binom{N-2}{k-2}\right)\varepsilon^2 + O(\varepsilon^4).\]

Maximum overlap over trajectory easy to calculate… with some caveats.

Given initial $\color{#090}\ket\phi$ and target $\color{#00c}\ket\psi$, what is the maximal $\lvert \color{#00c}\bra{\psi}\color{black}\underbrace{\exp(-i {\color{#d70}H} t){\color{#090}\ket{\phi}}}_{\color{#a00}\ket{\phi(t)}}\color{black}\rvert$? With $\color{#d70}H\color{black}=\sum_{\color{#770}k} e_{\color{#770}k} \color{#770}\ket{k}\bra{k}$, \[ \color{#00c}\langle\psi\color{black}\vert\color{#a00} \phi(t)\rangle\color{black}= \sum_{\color{#770}k} \color{#00c}\langle\psi\color{black}\vert\color{#770} k\rangle \langle k\color{black}\vert\color{#090} \phi\rangle \color{black}\exp(-i {\color{#770}e_k} t).\] Any phase structure $\exp(i{\color{#770}\alpha_k}) \approx \exp(-i {\color{#770}e_k} t)$ reachable $\implies$ \[\sup_t \lvert\color{#00c}\langle\psi\color{black}\vert\color{#a00} \phi(t)\rangle\color{black}\rvert = \sum_{\color{#770}k} \lvert\color{#00c}\langle\psi\color{black}\vert\color{#770} k\rangle \langle k\color{black}\vert\color{#090} \phi\rangle \color{black}\rvert .\]

All phases are reachable if eigenenergies are rationally independent.

Kronecker theorem:

Given $\vec e=(e_1, \ldots, e_n)\in\mathbb{R}^n$, the set \[\{(m e_1 \!\!\!\!\mod 1, m e_2\!\!\!\! \mod 1,\ldots) \in [0,1)^{n} : m \in\mathbb{Z}\}\] is dense in $[0,1)^n$ iff for $\vec q \in \mathbb{Q}$, $\vec e \cdot \vec q \not\in \mathbb{Q}$.

Q2: Are eigenenergies of generic $2$-body Hamiltonians rationally independent?

(if you're good at Galois theory let me know)

Semidefinite programming allows for calculation of admissible expectation values.

What is the minimal value of $\braket{X}_{\color{#a00}\phi(t)}$ for \[{\color{#a00}\ket{\phi(t)}}=\exp(-i {\color{#d70}H} t) \color{#090}\ket{\phi}?\] Relaxation (only moments have to agree): \[\left\{\begin{matrix}\min_{\rho} \operatorname{Tr} \rho X,\\\operatorname{Tr} {\color{#d70}H}^n \rho = \braket{{\color{#d70}H}^n}_{\color{#090}\phi},\\\rho\succcurlyeq0 \end{matrix}\right.\]
\[\text{SDP} \leftrightarrow \min_{\vec r\in\text{Spec}} \vec r \cdot \vec n\]

Polynomial theory enables validation of moment structure.

Q3: How to prove unreachability?
Given $\color{#00c}\ket\psi$, does there exist $\color{#090}\bigotimes_k \ket{\phi_k}$ and \[{\color{#d70}H}\color{black}=\sum_{i,j,\alpha,\beta} {\color{#909}\gamma_{i,j,\alpha,\beta}} \sigma^{(i)}_\alpha\sigma^{(j)}_\beta + \sum_{i,\alpha} {\color{#660}h_{i,\alpha}} \sigma^{(i)}_\alpha,\] such that moments of ${\color{#d70}H}$ coincide? \[\left\{\begin{matrix}\braket{{\color{#d70}H}}_{\color{#090}\phi}=\braket{{\color{#d70}H}}_{\color{#00c}\psi}\\\braket{{\color{#d70}H}^2}_{\color{#090}\phi}=\braket{{\color{#d70}H}^2}_{\color{#00c}\psi}\\\ldots\\\braket{{\color{#d70}H}^{2^N}}_{\color{#090}\phi}=\braket{{\color{#d70}H}^{2^N}}_{\color{#00c}\psi}\end{matrix}\right.\]
For product states $\color{#090}\bigotimes_k \ket{\phi_k}$ expectation values factorize: \[\Big\langle\prod_{i,\alpha_i} \sigma^{(i)}_{\alpha_i}\Big\rangle_{\color{#090}\phi}=\prod \color{#090}r_{\alpha_i}^{(i)}.\] Given $\color{#00c}\ket\psi$, do there exist real ${\color{#909}\gamma_{i,j,\alpha,\beta}}, {\color{#660}h_{i,\alpha}}, \color{#090}\vec r^{(k)}$ such that \[\left\{\begin{matrix}\overbrace{\braket{{\color{#d70}H}^n}_{\color{#090}\phi}}^{\text{poly}({\color{#909}\gamma}, {\color{#660}h}, {\color{#090}\vec r^{(k)}})}=\braket{{\color{#d70}H}^n}_\psi,\\ \color{#090}\vec r^{(k)}\cdot \vec r^{(k)}\color{black}=1?\end{matrix}\right.\]

Toy model: permutationally invariant states of $N$ qubits

  • $N+1$-dimensional space of $\ket\psi$ such that \[\operatorname{SWAP}_{i,j}\ket\psi=\ket\psi.\]
  • Product states: $\ket{\phi}^{\otimes N}$. ($\approx SU(2)$ coherent states)
  • 2-body dynamics spanned by \[\{J_{\alpha}=\sum\sigma^{(i)}_\alpha\}_{\alpha={x,y,z}}~~\text{and}~~\{J_\alpha J_\beta+J_\beta J_\alpha\}_{\alpha,\beta}.\]
  • Reachable states: 10 real parameters. (regardless of $N$)

Toy model: Semidefinite analysis of reachability

  • Random states seem unreachable.
  • But almost everything with some kind of structure is.
  • Read more: 
    https://quantumstat.es/note/
    /Reachable-correlations
Sets of $(\braket{X},\braket{Y})_\psi$ for all, coherent and reachable states: $N=23$ qubits, random $X$ and $Y$.

Toy model: unreachability proof

Let $\color{#090}\ket\phi = \ket{N,0}$ and $$\color{#00c} \ket{\psi}=\frac1{\sqrt2} \left(\ket{N-1,1}+\ket{1,N-1}\right).$$ Does there exist \[{\color{#d70}H}\color{black}=\sum_{\alpha,\beta} {\color{#909}\gamma_{\alpha,\beta}} \{J_\alpha,J_\beta\} + \sum_{\alpha} {\color{#909}\gamma_{\alpha}} J_\alpha,\] such that first two moments of ${\color{#d70}H}$ coincide? \[\left\{\begin{matrix}\braket{{\color{#d70}H}}_{\color{#090}\phi}=\braket{{\color{#d70}H}}_{\color{#00c}\psi}\\\braket{{\color{#d70}H}^2}_{\color{#090}\phi}=\braket{{\color{#d70}H}^2}_{\color{#00c}\psi}\end{matrix}\right.\]

No! (for $N\ge 7$)

  • The condition $\braket{{\color{#d70}H}}_{\color{#090}\phi}-\braket{{\color{#d70}H}}_{\color{#00c}\psi}=0$ $\equiv$ selected linear subspace of $\color{#909}\vec \gamma$.
  • On this subspace, $\braket{{\color{#d70}H^2}}_{\color{#090}\phi}-\braket{{\color{#d70}H^2}}_{\color{#00c}\psi}$ is positive definite.
    (strictly $>0$ for nonzero $\vec\gamma$)
  • If first condition holds, the second can't.

Promising results for $N$ qubits!

Summary

Several open questions:

  • Q1: Which states are unreachable with autonomous 2-body dynamics?
  • Q2: Are there hidden energy resonances in every 2-body Hamiltonian?
  • Q3: How to prove unreachability?
  • Q4: If $\ket\psi$ is reachable, is it reachable from any product state?

Thank you for your attention!

  1. H. Haselgrove, M. Nielsen, T. Osborne, Quantum states far from the energy eigenstates of any local Hamiltonian, PRL 2003.
    (the eigenstates of 2-body Hamiltonians do not cover the entire state space)
  2. N. Wyderka, F. Huber, O. Gühne, Constraints on correlations in multiqubit systems, PRA 2018.
    (odd-body (e.g. 3-body) Hamiltonians do have some additional conserved quantities
  3. B. Kraus, Local Unitary equivalence of multipartite pure states, PRL 2010.
    (verification if a given pure state is LU-equivalent to another)