Riemannian Line-Search on the Unitary Group: Bridging Gradient Descent and Quantum Signal Processing

Marek Gluza

NTU Singapore

slides.com/marekgluza

Tailoring polynomials for quantum signal processing using methods of Riemannian geometry

Marek Gluza

NTU Singapore

slides.com/marekgluza

Mathematically, states of the quantum computer are like arrows pointing from the center of the sphere to its surface.

Simple observation "Earth is not flat" leads to quantum algorithms. We will use that when we walk along of the equator, we think we are going straight but eventually we will wrap around it.

Fixing a direction and rotating the arrow, corresponds to a type of of quantum computing operation.

Riemannian geometry underlying quantum algorithms

On a flat surface DOWN-LEFT-UP-RIGHT will return to point of origin.

Riemannian geometry underlying quantum algorithms

On a flat surface DOWN-LEFT-UP-RIGHT will return to point of origin.

On a curved surface SOUTH-WEST-NORTH-EAST will spiral way.

South
West
East
North

Riemannian geometry underlying quantum algorithms

South
West
East
North

My work on double-bracket quantum algorithms shows how to use this spiraling effect to implement non-Euclidean gradient descent in quantum computing.

Regular machine learning fails for quantum computing but our generalization works. The 'failed' machine learning is still key for us - as a warm-start!

(Physical Review Letters '26)

Riemannian geometry underlying quantum algorithms

Riemannian geometry is essential for quantum computation 

  • The unitary group \(U(d)\) is a Riemannian manifold
  • It is an embedded manifold \(U(d) = \{M\in \mathbb C^{d\times d}:~M^{-1}=M^\dagger\}\)
  • The tangent space is \(\{W\in \mathbb C^{d\times d}:~W^{\dagger}= -W\} \simeq\{iH \mathrm{~where~} H=H^\dagger\in \mathbb C^{d\times d}\} \)
  • The geodesics are matrix exponentials \( \{e^{sW}\}_{s\in\mathbb R} \subset U(d)\) or \( \{e^{isH}\}_{s\in\mathbb R} \subset U(d)\)
  • Computing a "gradient" must output an element of the tangent space
\mathbb C^{d\times d}
U(d)
g
W
iH
g^\dagger = -g
e^{s g}
g
g^\dagger = -g

\(\partial_{i,j}\) points to the interior, not tangential

Keep this in mind for later: Unlike in flat space, these 4 steps spiral away from the point of origin

Riemannian geometry is essential for quantum computation 

  • The unitary group \(U(d)\) is a Riemannian manifold
  • The tangent space is \(\{H\in \mathbb C^{d\times d}:~H^{\dagger}= H\} \)
  • The geodesics are matrix exponentials \( \{e^{isH}\}_{s\in\mathbb R} \subset U(d)\)
  • \(U(d)\) is a curved manifold
e^{is A}e^{is B}e^{-is A}e^{-is B} \neq 1
e^{is A}
e^{isB}
e^{-isB}
e^{-is A}

Operating a quantum computer is all about the group of unitary matrices

\forall A=A^\dagger, B=B^\dagger: ~~M = e^{ [A,B]}\in U(d)

Think of rotations on a sphere

The Lie bracket of two 'velocities' is again a velocity:

Check: \([A,B]^\dagger = (AB- BA)^\dagger = B^\dagger A^\dagger -A^\dagger B^\dagger = -[A,B]\)

M^\dagger = (e^{ [A,B]})^\dagger = e^{ [A,B]^\dagger}=e^{ -[A,B]}=M^{-1}
U(d)
e^{ [A,B]}

\([A,B]^\dagger = -[A,B]\)

Check: \([A,B]^\dagger = -[A,B]\)

U(d) = \{ M\in \mathbb C^{d\times d}: M^{-1} = M^\dagger\}

Operating a quantum computer is all about the group of unitary matrices

\forall A=A^\dagger, B=B^\dagger: ~~M = e^{ [A,B]}\in U(d)

Think of rotations on a sphere

Fact 3: The Lie bracket of two 'velocities' is again a velocity 

U(d) = \{ M\in \mathbb C^{d\times d}: M^{-1} = M^\dagger\}
e^{isA}e^{isB}e^{-isA}e^{-isB} = e^{-s^2[A,B]}+O(s^3)
e^{is A}
e^{isB}
e^{-isB}
e^{-is A}

Operating a quantum computer is all about the group of unitary matrices

U(d) = \{ M\in \mathbb C^{d\times d}: M^{-1} = M^\dagger\}

Fact 3: The Lie bracket of two 'velocities' is again a velocity 

e^{isA}e^{isB}e^{-isA}e^{-isB} = e^{-s^2[A,B]}+O(s^3)
\forall A=A^\dagger, B=B^\dagger: ~~M = e^{ [A,B]}\in U(d)

The main tool of double-bracket quantum algorithms

e^{is A}
e^{isB}
e^{-isB}
e^{-is A}

[1] arxiv:2206.11772

[2] arxiv:2403.09187

[3] arxiv:2408.03987

[4] arxiv:2408.07431

[5] arxiv:2412.04554

[6] arxiv:2504.01065

[7] arxiv:2504.01077

[8] arxiv:2507.15065

[9] arxiv:2510.00302

E(\psi) = \langle \psi| H | \psi\rangle
| \psi\rangle \mapsto e^{is A}|\psi\rangle

Note that choosing \(A=H\) doesn't change the energy

E(\psi') = \langle \psi| e^{-is H} He^{isH} | \psi\rangle = E(\psi)

Let's find which directions \(A\) are more useful!

\(\partial_{i,j}\) points to the interior, not tangential so which direction is best?

g
W
iH
iA

Same for \(A=\ket{\psi}\bra{\psi}\) because

e^{is \ket\psi\bra\psi} | \psi\rangle = e^{is}\ket\psi

Riemannian gradient: Unique vector \(g\) in the tangent space
such that the directional derivative is a projection onto \(g\)

\nabla_W f(x) = \text{lim}_{s\rightarrow 0}\frac{ f(x+s W) - f(x)}{s} = \langle g, W\rangle

\(\partial_{i,j}\) points to the interior, not tangential so which direction is best?

g
W
iH
iA
Tr(ABC) = Tr(CAB)
\langle g, W\rangle_{\rm{HS}} = \mathrm{Tr}\left(g^\dagger W\right)
(|\psi\rangle\langle \psi|)^\dagger = |\psi\rangle\langle \psi|, H^\dagger = H

We will use very simple ingredients to find \(g\) for \(E(\psi) =  \langle \psi|  H | \psi\rangle\):

Hilbert-Schmidt scalar product:

Cyclicity of trace:

Tangent space:

Riemannian gradient: Unique vector \(g\) in the tangent space such that the directional derivative is a projection onto \(g\)

\nabla_W f(x) = \text{lim}_{s\rightarrow 0}\frac{ f(x+sW) - f(x)}{s} = \langle g, W\rangle
\nabla_A E(\psi) = \text{lim}_{s\rightarrow 0} \frac{ \langle \psi| e^{-isA}He^{isA}| \psi\rangle- \langle \psi|H | \psi\rangle}{s}
Tr(ABC) = Tr(CAB)\\ \Downarrow \\ Tr(|\psi\rangle\langle \psi| [H,A] ) = Tr([|\psi\rangle\langle \psi|,H]A)
E(\psi) = \langle \psi| H | \psi\rangle
| \psi\rangle \mapsto e^{is A}|\psi\rangle
=\partial_s \langle \psi| e^{-isA}He^{isA}| \psi\rangle_{|s=0}
= iTr\left[|\psi\rangle\langle\psi| [A,H]\right]
\nabla_W f(x) = \langle g, W\rangle
\nabla_A E(\psi) = \langle- i[|\psi\rangle\langle \psi|,H] , A\rangle_{\rm{HS}} \\\Downarrow\\ g \equiv -i[|\psi\rangle\langle \psi|,H]
|\psi'\rangle = e^{s[|\psi\rangle\langle \psi|,H]} |\psi\rangle
|\psi\rangle \mapsto e^{isg} |\psi\rangle
|\psi''\rangle = e^{s[|\psi'\rangle\langle \psi'|,H]} |\psi'\rangle

Double-bracket quantum algorithms:

Systematic framework for implementing  exponentials of commutators on quantum computers. This uncovered new unitary synthesis formulas.

Riemannian gradient: Unique vector \(g\) in the tangent space such that the directional derivative is a projection onto \(g\)

e^{is A}
e^{isB}
e^{-isB}
e^{-is A}

Double-bracket quantum algorithms

Click these links at slides.com/marekgluza

Diagonalization https://arxiv.org/abs/2206.11772
Imaginary-time evolution https://arxiv.org/abs/2412.04554
Quantum signal processing https://arxiv.org/abs/2504.01077
Grover's search https://arxiv.org/abs/2507.15065 Approximates ITE
U^\dagger H U = \begin{pmatrix} \lambda_0&&\\ &\lambda_1&\\ &&\ddots \end{pmatrix}
|\psi(\tau)\rangle = \frac{e^{-\tau H}|\psi\rangle}{\| e^{-\tau H}|\psi\rangle\|}
\frac{ \prod_{k=1}^K(H-z_k I)|\psi\rangle}{\| \prod_{k=1}^K(H-z_k I)|\psi\rangle\|}
\prod_{k=1}^Ke^{i\theta_k|\psi_k\rangle\langle\psi_k|} e^{s_k[|\psi_k\rangle\langle\psi_k|,H]}|\psi\rangle
H' = e^{-s[D,H]}H e^{s[D,H]}
e^{s[|\psi\rangle\langle\psi|,H]}|\psi\rangle +O(\tau^2)
(1-2|\psi\rangle\langle\psi|)\times e^{i t H_f}
e^{s[|\psi\rangle\langle\psi|,H]}|\psi\rangle

Exponentials of commutators solve the unitary synthesis problem in all these cases

Regular gradient descent

 

Secret sauce when \(x\) is big or \(f(x)\) is costly is the learning rate \(\eta\). Greedy gradient descent will use
\(\eta = \argmin_r f( x_k -r\nabla f(x_k)\))

 

Next, we do it 'quantumly':

 

Next, I will show you the optimal line-search solution for linear cost functions like \(C(\ket\psi) = \bra \psi H\ket\psi\)

\(s\) is the 'Riemannian' learning rate

\ket{\psi_{k}(s)} = e^{s[\ket{\psi_k}\bra{\psi_k},H]} \ket{\psi_k}
x_{k+1} = x_k -\eta \nabla f(x_k)
\ket{\psi_{k}(s)} = e^{s[\ket{\psi_k}\bra{\psi_k},H]} \ket{\psi_k}
s_k = \argmin_{s\in\mathbb R} \bra{\psi_{k}(s)} H \ket{\psi_{k}(s)}
min_{s\in\mathbb R} \bra{\psi_{k}(s)} H \ket{\psi_{k}(s)} = E_k - \sqrt{V_k }\left(\sqrt{1+\tfrac14S_k^2}-\tfrac12{S_k} \right)
S_k ={\bra{\Psi}(H-E_k)^3\ket{\Psi}}/{\sqrt{V_k}^3}

Y. Suzuki

V_k = \bra{\psi_k} H^2\ket{\psi_k}-E_k^2
E_k=\bra{\psi_k} H\ket{\psi_k}
E_{k+1}=E_k -\sqrt{V_k}

Energy filtering in quantum optimization

Task: Given Hermitian \(H\), prepare quantum computer in eigenvector \(|\lambda_0\rangle\) with smallest eigenvalue \(\lambda_0\).

 \(H = \sum_k \lambda_k |\lambda_k\rangle\langle \lambda_k|\)  which is braket notation for: If \(H v_k = \lambda_k v_k\) then \(H= \sum_k \lambda_k v_k^\top v_k\)

After decomposing \(\ket\psi = \sum_k \psi_k \ket{\lambda_k}\) we can filter energy by \(f(H)\ket\psi = \sum_k \psi_k f(\lambda_k) \ket{\lambda_k}\)

Note: We are optimizing \(E(\psi) = \bra\psi H\ket\psi\) and \( \bra{\lambda_0} H\ket{\lambda_0} = \lambda_0\) is the global minimizer

Imaginary-time evolution

 		 Non-unitary?
Quantum signal processing Probabilistic?
|\psi(\tau)\rangle = \frac{e^{-\tau H}|\psi\rangle}{\| e^{-\tau H}|\psi\rangle\|}
|\psi(z)\rangle = \frac{ p(H)|\psi\rangle}{\| p(H)|\psi\rangle\|}

\(f(\lambda) = e^{-\tau \lambda}\)

\(p(\lambda) = \) Lanczos polynomial

Energy filtering in quantum optimization

Task: Given Hermitian \(H\), prepare quantum computer in eigenvector \(|\lambda_0\rangle\) with smallest eigenvalue \(\lambda_0\).

 \(H = \sum_k \lambda_k |\lambda_k\rangle\langle \lambda_k|\)  which is braket notation for: If \(H v_k = \lambda_k v_k\) then \(H= \sum_k \lambda_k v_k^\top v_k\)

After decomposing \(\ket\psi = \sum_k \psi_k \ket{\lambda_k}\) we can filter energy by \(f(H)\ket\psi = \sum_k \psi_k f(\lambda_k) \ket{\lambda_k}\)

Note: We are optimizing \(E(\psi) = \bra\psi H\ket\psi\) and \( \bra{\lambda_0} H\ket{\lambda_0} = \lambda_0\) is the global minimizer

f(\tau) = 1-\tau H
E_{k+1}=E_k -\sqrt{V_k}
F_k = |\langle \psi| \lambda_k\rangle|^2
E_k =\sum_k F_k \lambda_k

Energy filtering in quantum optimization

Imaginary-time evolution

 		 Non-unitary?
Quantum signal processing Probabilistic?
|\psi(\tau)\rangle = \frac{e^{-\tau H}|\psi\rangle}{\| e^{-\tau H}|\psi\rangle\|}
|\psi(z)\rangle = \frac{ p(H)|\psi\rangle}{\| p(H)|\psi\rangle\|}

Task: Given Hermitian \(H\), prepare quantum computer in eigenvector \(|\lambda_0\rangle\) with smallest eigenvalue \(\lambda_0\).

\(f(\lambda) = e^{-\tau \lambda}\)

 \(H = \sum_k \lambda_k |\lambda_k\rangle\langle \lambda_k|\)  which is braket notation for: If \(H v_k = \lambda_k v_k\) then \(H= \sum_k \lambda_k v_k^\top v_k\)

\(p(\lambda) = \) Lanczos polynomial

After decomposing \(\ket\psi = \sum_k \psi_k \ket{\lambda_k}\) we can filter energy by \(f(H)\ket\psi = \sum_k \psi_k f(\lambda_k) \ket{\lambda_k}\)

Note: We are optimizing \(E(\psi) = \bra\psi H\ket\psi\) and \( \bra{\lambda_0} H\ket{\lambda_0} = \lambda_0\) is the global minimizer

Design choices for optimization

Diagonalization


 		 Inefficient?
Imaginary-time evolution
 		 Non-unitary?
Quantum signal processing Probabilistic?
U^\dagger H U = \begin{pmatrix} \lambda_0&&\\ &\lambda_1&\\ &&\ddots \end{pmatrix}
|\psi(\tau)\rangle = \frac{e^{-\tau H}|\psi\rangle}{\| e^{-\tau H}|\psi\rangle\|}
|\psi(z)\rangle = \frac{ p(H)|\psi\rangle}{\| p(H)|\psi\rangle\|}
\ket{\psi_{k}(s)} = e^{s[\ket{\psi_k}\bra{\psi_k},H]} \ket{\psi_k}
min_{s\in\mathbb R} \bra{\psi_{k}(s)} H \ket{\psi_{k}(s)} = E_k - \sqrt{V_k }\left(\sqrt{1+\tfrac14S_k^2}-\tfrac12{S_k} \right)
S_k =\frac{\bra{\Psi}(H-E_k)^3\ket{\Psi}}{\sqrt{V_k}^3}
V_k = \bra{\psi_k} H^2\ket{\psi_k}-E_k^2
E_k=\bra{\psi_k} H\ket{\psi_k}
s_k = \frac{-\arctan(\frac{2}{S_k})}{2\sqrt{V_k}}

Greedy optimal line-search iteration: 

min_{s\in\mathbb R} \bra{\psi_{k}(s)} H \ket{\psi_{k}(s)} = E_k - \sqrt{V_k }\left(\sqrt{1+\tfrac14S_k^2}-\tfrac12{S_k} \right)

Around step \(k=3\) a global method tempers being greedy.

Then the global method catches up by having a larger \(V_k\) in the subsequent steps.

s_k = \frac{-\arctan(\frac{2}{S_k})}{2\sqrt{V_k}}

Greedy optimal line-search iteration: 

min_{s\in\mathbb R} \bra{\psi_{k}(s)} H \ket{\psi_{k}(s)} = E_k - \sqrt{V_k }\left(\sqrt{1+\tfrac14S_k^2}-\tfrac12{S_k} \right)

Geometrically, this means there was a 'ridge' that later eases the climb.

s_k = \frac{-\arctan(\frac{2}{S_k})}{2\sqrt{V_k}}

Greedy optimal line-search iteration: 

min_{s\in\mathbb R} \bra{\psi_{k}(s)} H \ket{\psi_{k}(s)} = E_k - \sqrt{V_k }\left(\sqrt{1+\tfrac14S_k^2}-\tfrac12{S_k} \right)

Next, I will tell you that \(k\) steps of Riemannian gradient descent are quantum signal processing by a polynomial of degree \(k\).

 

This means we can compare to the ideal polynomial the "Lanczos polynomial".

Energy filtering in quantum optimization

Imaginary-time evolution

 		 Non-unitary?
Quantum signal processing Probabilistic?
|\psi(\tau)\rangle = \frac{e^{-\tau H}|\psi\rangle}{\| e^{-\tau H}|\psi\rangle\|}
|\psi(z)\rangle = \frac{ p(H)|\psi\rangle}{\| p(H)|\psi\rangle\|}

Task: Given Hermitian \(H\), prepare quantum computer in eigenvector \(|\lambda_0\rangle\) with smallest eigenvalue \(\lambda_0\).

\(f(\lambda) = e^{-\tau \lambda}\)

 \(H = \sum_k \lambda_k |\lambda_k\rangle\langle \lambda_k|\)  which is braket notation for: If \(H v_k = \lambda_k v_k\) then \(H= \sum_k \lambda_k v_k^\top v_k\)

\(p(\lambda) = \) Lanczos polynomial

After decomposing \(\ket\psi = \sum_k \psi_k \ket{\lambda_k}\) we can filter energy by \(f(H)\ket\psi = \sum_k \psi_k f(\lambda_k) \ket{\lambda_k}\)

Note: We are optimizing \(E(\psi) = \bra\psi H\ket\psi\) and \( \bra{\lambda_0} H\ket{\lambda_0} = \lambda_0\) is the global minimizer

Energy filtering in quantum optimization

Imaginary-time evolution

 		 Non-unitary?
Quantum signal processing Probabilistic?
|\psi(\tau)\rangle = \frac{e^{-\tau H}|\psi\rangle}{\| e^{-\tau H}|\psi\rangle\|}
|\psi(z)\rangle = \frac{ p(H)|\psi\rangle}{\| p(H)|\psi\rangle\|}

Task: Given Hermitian \(H\), prepare quantum computer in eigenvector \(|\lambda_0\rangle\) with smallest eigenvalue \(\lambda_0\).

\(f(\lambda) = e^{-\tau \lambda}\)

 \(H = \sum_k \lambda_k |\lambda_k\rangle\langle \lambda_k|\)  which is braket notation for: If \(H v_k = \lambda_k v_k\) then \(H= \sum_k \lambda_k v_k^\top v_k\)

\(p(\lambda) = \) Lanczos polynomial

After decomposing \(\ket\psi = \sum_k \psi_k \ket{\lambda_k}\) we can filter energy by \(f(H)\ket\psi = \sum_k \psi_k f(\lambda_k) \ket{\lambda_k}\)

Note: We are optimizing \(E(\psi) = \bra\psi H\ket\psi\) and \( \bra{\lambda_0} H\ket{\lambda_0} = \lambda_0\) is the global minimizer

|\psi'\rangle = \frac{P(H) |\psi\rangle}{\|P(H) |\psi\rangle\|} = U_\psi(P) |\psi\rangle
U_\psi(P)
|\phi'\rangle = e^{s[|\phi\rangle\langle\phi|,H]} |\phi\rangle
U_\phi(P) = e^{s[|\phi\rangle\langle\phi|,H]}

We will show that 

\(P(H) = 1-\tau_sH\)

Ansatz:

|\psi'\rangle = e^{s[|\psi\rangle\langle \psi|,H]} |\psi\rangle
|\psi'\rangle = \sum_{n=0}^\infty \frac{s^n}{n!} ([|\psi\rangle\langle \psi|,H])^n |\psi\rangle
[|\psi\rangle\langle \psi|,H] |\psi\rangle = |\psi\rangle\langle \psi|H |\psi\rangle - H|\psi\rangle\langle \psi |\psi\rangle
[|\psi\rangle\langle \psi|,H] |\psi\rangle = (E-H) |\psi\rangle

Double-bracket ansatz:

\(n=1\):

([|\psi\rangle\langle \psi|,H])^2 |\psi\rangle = [|\psi\rangle\langle \psi|,H](E-H) |\psi\rangle
([|\psi\rangle\langle \psi|,H])^2 |\psi\rangle = E(E-H) |\psi\rangle - [|\psi\rangle\langle \psi|,H] H |\psi\rangle
= - V|\psi\rangle
([|\psi\rangle\langle \psi|,H])^{2k} |\psi\rangle = (- 1)^k V^k|\psi\rangle
([|\psi\rangle\langle \psi|,H])^{2k+1} |\psi\rangle = (- 1)^k V^k(E-H)|\psi\rangle
[|\psi\rangle\langle \psi|,H] |\psi\rangle = (E-H) |\psi\rangle

\(n=1\):

\(n=2\):

Y. Suzuki

= E^2 |\psi\rangle - EH |\psi\rangle - |\psi\rangle\langle \psi|H^2 |\psi\rangle + E H|\psi\rangle
V =\langle \psi |H^2|\psi\rangle - \langle \psi |H|\psi\rangle^2
([|\psi\rangle\langle \psi|,H])^{2k} |\psi\rangle = (- 1)^k V^k|\psi\rangle
([|\psi\rangle\langle \psi|,H])^{2k+1} |\psi\rangle = (- 1)^k V^k(E-H)|\psi\rangle

This starts looking like quantum signal processing:

Lemma 1. Any linear polynomial with a real root \(P(H) = 1-\tau H\) can be implemented this way.

 

https://arxiv.org/abs/2504.01077

Z. Holmes

Y. Suzuki

B. Tiang

J. Son

N. Ng

https://arxiv.org/abs/2504.01077

s_k = \frac{-\arctan(\frac{2}{S_k})}{2\sqrt{V_k}}

Greedy optimal line-search iteration: 

min_{s\in\mathbb R} \bra{\psi_{k}(s)} H \ket{\psi_{k}(s)} = E_k - \sqrt{V_k }\left(\sqrt{1+\tfrac14S_k^2}-\tfrac12{S_k} \right)

arxiv.org/abs/2208.00567

There is a very nice method for finding the Lanczos polynomial from block-encodings

Being greedy makes you globally slower

Being greedy makes you more stable under noise!

arxiv.org/abs/2208.00567

Full compilation in Qrisp of GOLS and KMM

Checking GOLS with MPS

New solver for Krylov spaces:

Iterating OLS to converge to Lanczos by removing random roots and finding new ones

E_{k+1} \ge E_k - \sqrt{V_k }\left(\sqrt{1+\tfrac14S_k^2}-\tfrac12{S_k} \right)

On Arxiv soon-ish

Full compilation in Qrisp of GOLS and KMM

Checking GOLS with MPS

New solver for Krylov spaces:

Iterating OLS to converge to Lanczos by removing random roots and finding new ones

E_{k+1} \ge E_k - \sqrt{V_k }\left(\sqrt{1+\tfrac14S_k^2}-\tfrac12{S_k} \right)

On Arxiv soon-ish

4 stages of creating quantum algorithms

1. Design choice:
How to go about it?

|\psi(\tau)\rangle = \frac{e^{-\tau H}|\psi\rangle}{\| e^{-\tau H}|\psi\rangle\|}

2. Unitary synthesis:
How to do it?

3. Circuit compilation:
What gates to do?

0. Problem choice:
What challenge to take up?

Double-bracket quantum algorithms:

Systematic framework for unitary synthesis

4 stages of creating quantum algorithms

1. Design choice:
How to go about it?

|\psi(\tau)\rangle = \frac{e^{-\tau H}|\psi\rangle}{\| e^{-\tau H}|\psi\rangle\|}

3. Circuit compilation:
What gates to do?

0. Problem choice:
What challenge to take up?

2. Unitary synthesis:
How to do it?

Double-bracket quantum algorithms:

Systematic framework for unitary synthesis

4 stages of creating quantum algorithms

E(\psi) = \langle \psi| H | \psi\rangle

4 stages of creating quantum algorithms

1. Design choice:
How to go about it?

|\psi(\tau)\rangle = \frac{e^{-\tau H}|\psi\rangle}{\| e^{-\tau H}|\psi\rangle\|}

3. Circuit compilation:
What gates to do?

|\psi(\tau)\rangle = \frac{e^{-\tau H}|\psi\rangle}{\| e^{-\tau H}|\psi\rangle\|}

0. Problem choice:
What challenge to take up?

2. Unitary synthesis:
How to do it?

4 stages of creating quantum algorithms

Imaginary-time evolution

\ket\psi\rightarrow |\psi(\tau)\rangle
H = \sum_{k=1}^{D} \lambda_k \ket{\lambda_k}\bra{\lambda_k}
e^{-\tau H}|\psi\rangle = \langle\lambda_0|\psi\rangle \ket{\lambda_0} +\sum_{k=1}^{D} e^{-\tau\lambda_k} \langle\lambda_k|\psi\rangle \ket{\lambda_k}
|\psi\rangle = \langle\lambda_0|\psi\rangle \ket{\lambda_0} +\sum_{k=1}^{D} \langle\lambda_k|\psi\rangle \ket{\lambda_k}
\lim_{\tau\rightarrow \infty}\ket{\psi(\tau)} = \ket{\lambda_0}

4 stages of creating quantum algorithms

1. Design choice:
How to go about it?

|\psi(\tau)\rangle = \frac{e^{-\tau H}|\psi\rangle}{\| e^{-\tau H}|\psi\rangle\|}

3. Circuit compilation:
What gates to do?

|\psi(\tau)\rangle = \frac{e^{-\tau H}|\psi\rangle}{\| e^{-\tau H}|\psi\rangle\|}

0. Problem choice:
What challenge to take up?

2. Unitary synthesis:
How to do it?

|\psi(\tau)\rangle = \frac{e^{-\tau H}|\psi\rangle}{\| e^{-\tau H}|\psi\rangle\|} = e^{\tau[|\psi\rangle\langle\psi|,H]}|\psi\rangle +O(\tau^2)

4 stages of creating quantum algorithms

4 stages of creating quantum algorithms

1. Design choice:
How to go about it?

|\psi(\tau)\rangle = \frac{e^{-\tau H}|\psi\rangle}{\| e^{-\tau H}|\psi\rangle\|}

3. Circuit compilation:
What gates to do?

0. Problem choice:
What challenge to take up?

2. Unitary synthesis:
How to do it?

Double-bracket quantum algorithms:

Systematic framework for unitary synthesis

4 stages of creating quantum algorithms

E(\psi) = \langle \psi| H | \psi\rangle

Operating a quantum computer is all about the group of unitary matrices

\forall A=A^\dagger, B=B^\dagger: ~~M = e^{ [A,B]}\in U(d)

Fact 3: The Lie bracket of two 'velocities' is again a velocity 

U(d) = \{ M\in \mathbb C^{d\times d}: M^{-1} = M^\dagger\}
e^{isA}e^{isB}e^{-isA}e^{-isB} = e^{-s^2[A,B]}+O(s^3)
e^{is A}
e^{isB}
e^{-isB}
e^{-is A}

Operating a quantum computer is all about the group of unitary matrices

U(d) = \{ M\in \mathbb C^{d\times d}: M^{-1} = M^\dagger\}

Fact 3: The Lie bracket of two 'velocities' is again a velocity 

e^{isA}e^{isB}e^{-isA}e^{-isB} = e^{-s^2[A,B]}+O(s^3)
\forall A=A^\dagger, B=B^\dagger: ~~M = e^{ [A,B]}\in U(d)

The main tool of double-bracket quantum algorithms

e^{is A}
e^{isB}
e^{-isB}
e^{-is A}

4 stages of creating quantum algorithms

1. Design choice:
How to go about it?

|\psi(\tau)\rangle = \frac{e^{-\tau H}|\psi\rangle}{\| e^{-\tau H}|\psi\rangle\|}

3. Circuit compilation:
What gates to do?

0. Problem choice:
What challenge to take up?

2. Unitary synthesis:
How to do it?

Double-bracket quantum algorithms:

Systematic framework for unitary synthesis

4 stages of creating quantum algorithms

|\psi(\tau)\rangle = \frac{e^{-\tau H}|\psi\rangle}{\| e^{-\tau H}|\psi\rangle\|}
E(\psi) = \langle \psi| H | \psi\rangle
e^{s[|\psi\rangle\langle \psi|,H]} |\psi\rangle

4 stages of creating quantum algorithms

\frac{e^{-\tau H}|\psi\rangle}{\| e^{-\tau H}|\psi\rangle\|} = e^{\tau[|\psi\rangle\langle\psi|,H]}|\psi\rangle +O(\tau^2)

Product formula approximation:

\(e^{\tau[|\psi\rangle\langle\psi|,H]} =  e^{i\sqrt{\tau}H}e^{i\sqrt{\tau}|\psi\rangle\langle\psi|}e^{-i\sqrt{\tau}H}e^{-i\sqrt{\tau}|\psi\rangle\langle\psi|} + O(\tau^{3/2})\)

4 stages of creating quantum algorithms

2. Unitary synthesis:
How to do it?

1. Design choice:
How to go about it?

|\psi(\tau)\rangle = \frac{e^{-\tau H}|\psi\rangle}{\| e^{-\tau H}|\psi\rangle\|}

3. Circuit compilation:
What gates to do?

|\psi(\tau)\rangle = \frac{e^{-\tau H}|\psi\rangle}{\| e^{-\tau H}|\psi\rangle\|}

0. Problem choice:
What challenge to take up?

= e^{i\sqrt{\tau}H}e^{i\sqrt{\tau}|\psi\rangle\langle\psi|}e^{i\sqrt{\tau}H}|\psi\rangle + O(\tau^{3/2})

3. Circuit compilation:
What gates to do?

Product formula approximation:

\(e^{\tau[|\psi\rangle\langle\psi|,H]} =  e^{i\sqrt{\tau}H}e^{i\sqrt{\tau}|\psi\rangle\langle\psi|}e^{-i\sqrt{\tau}H}e^{-i\sqrt{\tau}|\psi\rangle\langle\psi|} + O(\tau^{3/2})\)

Quantum algorithm DB-QITE - iterate recursively:

  1. Define \(|{\psi_k}\rangle = U_k |0\rangle\)
  2. Use \(e^{is |\psi_k\rangle\langle\psi_k|} = U_ke^{is |0\rangle\langle0|}U_k^\dagger \)
  3. Recursively iterate \( U_{k+1} = e^{is H} U_k e^{is |0\rangle\langle 0|} U_k^\dagger e^{-is H} U_k\)

3 stages of creating quantum algorithms

1. Design choice:
How to go about it?

3. Circuit compilation:
What gates to do?

|\psi(\tau)\rangle = \frac{e^{-\tau H}|\psi\rangle}{\| e^{-\tau H}|\psi\rangle\|}

0. Problem choice:
What challenge to take up?

e^{itH}\\ e^{it|\psi\rangle\langle\psi|}

4 stages of creating quantum algorithms

2. Unitary synthesis:
How to do it?

(PRL '26)

3. Circuit compilation:
What gates to do?

Product formula approximation:

\(e^{\tau[|\psi\rangle\langle\psi|,H]} =  e^{i\sqrt{\tau}H}e^{i\sqrt{\tau}|\psi\rangle\langle\psi|}e^{-i\sqrt{\tau}H}e^{-i\sqrt{\tau}|\psi\rangle\langle\psi|} + O(\tau^{3/2})\)

Quantum algorithm DB-QITE - iterate recursively:

  1. Define \(|{\psi_k}\rangle = U_k |0\rangle\)
  2. Use \(e^{is |\psi_k\rangle\langle\psi_k|} = U_ke^{is |0\rangle\langle0|}U_k^\dagger \)
  3. Recursively iterate \( U_{k+1} = e^{is H} U_k e^{is |0\rangle\langle 0|} U_k^\dagger e^{-is H} U_k\)

3 stages of creating quantum algorithms

1. Design choice:
How to go about it?

3. Circuit compilation:
What gates to do?

|\psi(\tau)\rangle = \frac{e^{-\tau H}|\psi\rangle}{\| e^{-\tau H}|\psi\rangle\|}

0. Problem choice:
What challenge to take up?

e^{itH}\\ e^{it|\psi\rangle\langle\psi|}

4 stages of creating quantum algorithms

2. Unitary synthesis:
How to do it?

(accepted at PRL)

Numerical results for DB-QITE:

DB-QITE:

  1. Define \(|{\psi_k}\rangle = U_k |0\rangle\)
  2. Recursively iterate \( U_{k+1} = e^{is H} U_k e^{is |0\rangle\langle 0|} U_k^\dagger e^{-is H} U_k\)

Then:







https://arxiv.org/abs/2412.04554

Quantinuum

(accepted at PRL)

https://www.meteo.pl/um/php/pict_show.php?cat=2&time=7&date=2022072418

https://www.quantamagazine.org/the-era-of-quantum-computing-is-here-outlook-cloudy-20180124/

Quantum computing: What do we really have?

Yes. But: Is it really enough?

Quantum winter

Quantum computers

Useful tasks?

If for
No
Then
BUT!

Quantum winter

Quantum computers

Useful tasks!

If for
No
Then

Material science?

\langle\hat \phi_k^2(t)\rangle= \cos^2(\omega_k t) \langle \phi_k^2(0)\rangle \\ \quad\quad\quad\quad\quad+\sin^2(\omega_k t) \langle \delta\hat \rho_k^2(0)\rangle
\delta \hat\rho
\langle\hat \phi^2(t)\rangle
\langle\delta\hat \rho^2(0)\rangle =0?
\langle \hat \phi^2(0)\rangle
\hat \phi

Gaussian quantum simulators

https://arxiv.org/abs/2004.04174

https://arxiv.org/abs/2201.00049

https://arxiv.org/abs/1101.2659

https://arxiv.org/abs/2003.01808

Ask me anytime:

 

[4,8]

[1]

[3]

[2]

[5]

[13]

[6]

[9]

[11,16]

[12]

[7, 14, 15, 17]

[10]

arXiv:2206.11772

0

0

0

0

C

Fidelity witnesses

Tomography optical lattices

Tomography phonons

Proving statistical mechanics

Quantum simulating DSF

Holography in tensor networks

PEPS contraction average #P-hard

Quantum field machine

MBL l-bits

Gaussian quantum simulators

https://arxiv.org/abs/2004.04174

https://arxiv.org/abs/2201.00049

https://arxiv.org/abs/1101.2659

https://arxiv.org/abs/2003.01808

Gaussian quantum simulators

https://arxiv.org/abs/2004.04174

https://arxiv.org/abs/2201.00049

https://arxiv.org/abs/1101.2659

https://arxiv.org/abs/2003.01808

Gaussian quantum simulators

https://arxiv.org/abs/2004.04174

https://arxiv.org/abs/2201.00049

https://arxiv.org/abs/1101.2659

https://arxiv.org/abs/2003.01808

How?

Ultra-cold 1d gases

Inside: atoms

Outside:  wavepackets

hydrodynamics

 

Energy of phonons

E
E
E
E
\hat H_\text{TLL} = \int \text{d}z\left[ \frac{\hbar^2n_{GP}}{2m}\partial_z\hat\varphi^2+g\delta\hat\varrho^2\right]
\delta\varrho
\partial_z\varphi
\partial_z\varphi
\ll
\delta\varrho
\ll

Tomonaga-Luttinger liquid

Quantum field refrigerators in the TLL model:

System

Piston

Bath

Bath with excitations

System cooled down

 Breaking of the Huygens-Fresnel principle

 in the inhomogenous TLL model:

https://arxiv.org/abs/2104.07751

https://arxiv.org/abs/2006.01177

Why?

Why develop continuous field

quantum simulators?

  • Representation theory: Quantum information?
  • Continuum limits: BQP and QMA or more?
  • Are nanowires computationally hard to simulate?

What do we know is difficult?

SM

Fundamental

Universal

Effective

Why develop continuous field

quantum simulators?

  • Representation theory: Quantum information?
  • Continuum limits: BQP and QMA or more?
  • Are nanowires computationally hard to simulate?

What do we know is difficult?

SM

Fundamental

Universal

Effective

Non-thermal

steady states

Sine-Gordon

thermal states

Atomtronics

Generalized hydrodynamics

Recurrences

Some highlights:

Schweigler, Schmiedmayer,

et. al, arxiv:1505.03126

Langen, Schmiedmayer,

et. al, arxiv:1411.7185

Rauer, Schmiedmayer,

et. al, arxiv:1705.08231

Schweigler, Schmiedmayer,

et. al, arxiv:2003.01808

Aidelsburger, Beugnon,

et. al, arXiv:1705.02650

Eckel, Campbell,

et. al, arXiv:1406.1095

Møller, Schmiedmayer,

et. al,  arXiv:2006.08577

Malvania, Weiss,

et. al, arxiv:2009.06651

Schemmer, Bouchoule,

et. al, arxiv:1810.07170

Interferometry measures velocities

van Nieuwkerk, Schmiedmayer, Essler, arXiv:1806.02626

Schumm, Schmiedmayer, Kruger, et al., arXiv:quant-ph/0507047

\partial_z\hat \varphi(z) \approx\frac{\hat \varphi(z)-\hat \varphi(z+\Delta z)}{\Delta z}
\partial_z\hat \varphi(z) \approx\frac{\Delta\hat\varphi(z)}{\Delta z}
\Delta z

Tomography

Tomography for phonons

\hat H_\text{TLL} = \int \text{d}z\left[ \frac{\hbar^2n_{GP}}{2m}\partial_z\hat\varphi^2+g\delta\hat\varrho^2\right]
\hat H_\text{TLL} = \sum_{k>0} \frac {\hbar \omega_k}2 (\phi_k^2+\delta\hat\rho_k^2) + g\hat\rho_0^2
\langle\hat \phi_k^2(t)\rangle= \cos^2(\omega_k t) \langle \phi_k^2(0)\rangle \\ \quad\quad\quad\quad\quad+\sin^2(\omega_k t) \langle \delta\hat \rho_k^2(0)\rangle
\delta \hat\rho
\langle\hat \phi^2(t)\rangle
\langle\delta\hat \rho^2(0)\rangle =0?
\langle \hat \phi^2(0)\rangle
\hat \phi

Tomography for phonons

\hat H_\text{TLL} = \int \text{d}z\left[ \frac{\hbar^2n_{GP}}{2m}\partial_z\hat\varphi^2+g\delta\hat\varrho^2\right]
\hat H_\text{TLL} = \sum_{k>0} \frac {\hbar \omega_k}2 (\phi_k^2+\delta\hat\rho_k^2) + g\hat\rho_0^2
\langle\hat \phi_k^2(t)\rangle= \cos^2(\omega_k t) \langle \phi_k^2(0)\rangle \\ \quad\quad\quad\quad\quad+\sin^2(\omega_k t) \langle \delta\hat \rho_k^2(0)\rangle
\delta \hat\rho
\langle\hat \phi^2(t)\rangle
\langle\delta\hat \rho^2(0)\rangle =0?
\langle \hat \phi^2(0)\rangle
\hat \phi

https://arxiv.org/abs/2108.07829

https://arxiv.org/abs/1807.04567

https://arxiv.org/abs/2005.09000

What are eigenmodes?

\hat H_\text{TLL} = \int \text{d}z\left[ \frac{\hbar^2n_{GP}}{2m}\partial_z\hat\varphi^2+g\delta\hat\varrho^2\right]
\hat H_\text{TLL} = \sum_{k>0} \frac {\hbar \omega_k}2 (\phi_k^2+\delta\hat\rho_k^2) + g\hat\rho_0^2
\hat \phi_k = \int_0^L dz \cos(\pi k z/L)\hat \varphi(z)
\int_0^L dz \cos(\pi k z/L)\cos(\pi k'z/L) = \delta_{k,k'}

https://arxiv.org/abs/2104.07751

Transmutation

\langle\hat \phi_k^2(t)\rangle= \cos^2(\omega_k t) \langle \hat\phi_k^2(0)\rangle +\sin^2(\omega_k t) \langle \delta\hat \rho_k^2(0)\rangle
\langle\hat \phi_k^2(t)\rangle= \langle \hat\phi_k^2(0)\rangle
\langle\hat \phi_k^2(t)\rangle= \langle \delta\hat \rho_k^2(0)\rangle
t=0:
t=\frac{\pi}{2\omega_k}:

https://arxiv.org/abs/2108.07829

\delta \hat\rho
\langle\hat \phi^2(t)\rangle
\langle\delta\hat \rho^2(0)\rangle =0?
\langle \hat \phi^2(0)\rangle
\hat \phi

Tomography

A_{1,2}= \sin^2(\omega_k t_1 )
A_{1,1}= \cos^2(\omega_k t_1)
\langle\hat \phi_k^2(t)\rangle= \cos^2(\omega_k t) \langle \hat\phi_k^2(0)\rangle+ \sin^2(\omega_k t) \langle \delta\hat \rho_k^2(0)\rangle
A_{3,2}= \sin^2(\omega_k t_3 )
A_{3,1}= \cos^2(\omega_k t_3)
A_{2,2}= \sin^2(\omega_k t_2 )
A_{2,1}= \cos^2(\omega_k t_2)
A \begin{pmatrix} \langle \hat\phi_k^2(0)\rangle \\\langle \delta\hat \rho_k^2(0)\rangle \end{pmatrix} =\begin{pmatrix} \langle \hat\phi_k^2(t_1)\rangle \\\langle \hat\phi_k^2(t_2)\rangle \\\langle \hat\phi_k^2(t_3)\rangle \end{pmatrix}
\|Aq - d\| = \text{min}

(This formalism: Tomography for many modes)

Tomography Klein-Gordon thermal state after quench

\hat H_\text{KG}=\hat H_\text{TLL}+J\int \mathrm{d}z \,n_{GP} \hat \varphi^2

https://arxiv.org/abs/2108.07829

https://arxiv.org/abs/1807.04567

Extracting physical properties

https://arxiv.org/abs/1807.04567

Extracting physical properties

https://arxiv.org/abs/1807.04567

https://arxiv.org/abs/1705.08231

Extracting physical properties

https://arxiv.org/abs/1807.04567

Tomography for optical lattices

https://arxiv.org/abs/2005.09000

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