Quantum algorithms for optimisation derived from the non-Euclidean geometry of quantum computing operations

Marek Gluza

NTU Singapore

Scan QR code or go to slides.com/marekgluza

What is a quantum computer?

It is a quantum system e.g. ionized atom, loop of a superconductor

It is modeled by a finite dimensional vector space $\mathcal V = \mathbb C^d$

It is modular: We can add more of its components and make $d$ grow

It is programmable: There are ways (quantum gates) to change its states in a reproducible and reversible manner

It is universal: Composing sufficiently many of the programmable operations allows to approximate any $v = |v\rangle \in \mathcal V$

We take more qubits

Apply patterned EM radiation
Reproducible: Use FPGA

Reversible: Physics challenge; keep cold

Physics papers from the 90's

We can make it!

We take $n$ qubits then $\mathcal V = (\mathbb C^2)^{\otimes n}$ and $d=2^n$

What is a quantum computer?

A quantum computer is an experimental setup A which includes a quantum system B and that setup A allows to manipulate the quantum state of B.

This is modeled by mulitplying unitary matrices to vectors from a finite-dimensional vector space.

Types of quantum computers:

A) Superconducting loops
B) Ionized atoms
C) Neutral atoms
D) Others

Companies:

A) Google, IQM, IBM, Rigetti
B) Quantinuum, IonQ
C) Quera, Atom Computing, Pasqal
D) Alice&Bob, Qolab, Normal Computing, Psi-Quantum, ~~Zapata~~

Currently 80 companies listed here:

https://thequantuminsider.com/2025/09/23/top-quantum-computing-companies/

What are quantum algorithms?

We model quantum computing operations by mulitplying unitary matrices to vectors from a finite-dimensional vector space:

States of the quantum computer are vectors.
Operations on the quantum computer rotate these vectors.
Two consecutive rotations are a rotation.

The space of all rotations generalizes the geometry of the sphere.
Straight lines in the space of rotations are those which don't change their direction.
Going SOUTH-WEST-NORTH-EAST in the space of all rotations spirals away from the point of origin.
Repeating two directions sufficiently many times will bring you eventually everywhere.

Braket notation

|\psi\rangle = a | 0 \rangle + b | 1\rangle

\langle \vec v,\vec w\rangle \leftrightarrow \langle \phi|\psi\rangle

1 qubit:

*The biography of the physicist who proposed it is literally called "The strangest man"

\mathcal V = \mathbb C^2 = \mathrm{span}_{\mathbb C}\{ e_1, e_2\}

| 0 \rangle :=e_1, | 1\rangle :=e_2

v \in \mathcal V \leftrightarrow |v\rangle = v_0 | 0 \rangle + v_1 | 1\rangle

\langle \vec v,A \vec v\rangle \leftrightarrow \langle \psi|H|\psi\rangle

A \vec v \leftrightarrow H|\psi\rangle

*Dirac was one of the smartest quantum physicist in history

\vec v^\dagger \in \mathcal V^* \leftrightarrow \langle \psi|

"ket"

"bra"

"bracket"

"quantum superposition"

What is a universal quantum computer?

|\psi\rangle = a | 0 \rangle + b | 1\rangle

1 qubit

2 qubits

3 qubits

4 qubits

|\psi\rangle = U |00\ldots0\rangle \Rightarrow U \approx U_c

it can approximate any state preparation via circuits

It is a machine for realizing such linear combinations in nature

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

M,M' \in U(d) \Rightarrow (M M')(MM')^\dagger = M (M'M'^\dagger) M^\dagger

= M M^\dagger = I

\Rightarrow (M M')\in U(d)

\ket{\psi'} = M \ket\psi \Rightarrow \ket{\psi''} = M'\ket{\psi'} = M'M \ket\psi

Think of rotations on a sphere

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

|\psi'\rangle

|\psi\rangle

|\psi''\rangle

|\psi\rangle = U |00\ldots0\rangle

What are quantum algorithms?

We model quantum computing operations by mulitplying unitary matrices to vectors from a finite-dimensional vector space:

States of the quantum computer are vectors.
Operations on the quantum computer rotate these vectors.
Two consecutive rotations are a rotation.
The space of all rotations generalizes the geometry of the sphere.
Straight lines in the space of rotations are those which don't change their direction.
Going SOUTH-WEST-NORTH-EAST in the space of all rotations spirals away from the point of origin.
Repeating two directions sufficiently many times will bring you eventually everywhere.

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^\dagger = -g

e^{s g}

g^\dagger = -g

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

To go straight to New York, follow the equator. Don't drill a tunnel to literally go straight.

Unlike in flat space, going SOUTH-WEST-NORTH-EAST in the space of all rotations spirals 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

\exist G_1,G_2,\ldots G_K \in U(d) \mathrm{~such~that}

Think of rotations on a sphere

Universal quantum computation means: Composing sufficiently many basic unitary matrices (elementary quantum gates $G_1,G_2,\ldots, G_K$) can approximate any other 'larger' unitary.

(finding these sequences = task of unitary synthesis)

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

\forall M \in U(d)~\exist \mathrm{sequence~s.t.} ~M\approx G_1 G_5G_6G_1 G_K G_1 G_2\ldots

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

Quantum computing cannot be useful if

The problem can be solved quickly on classical computers
Solving the problem isn't important
Solving the problem is too hard for the quantum computer

Classical computation

Key question in current quantum computation:
Find an important problem which is difficult yet doable

Millenials

What challenge to take up?

Quantum computing cannot be useful if

The problem can be solved quickly on classical computers
Solving the problem isn't important
Solving the problem is too hard for the quantum computer

Key question in current quantum computation:
Find an important problem which is difficult yet doable

What challenge to take up?

Materials science?

Why materials science? Imagine improving photovoltaics by 1%.

What needs to be done? Perform $\bra\psi H\ket\psi$ optimization.

How? Quantum circuit synthesis.

Non-Euclidean geometry leads to quantum algorithms

Next: How to get $$ from a quantum computer?

We need to optimize among all operations of the quantum computer to get a low-energy state

(Non-Euclidean optimization)

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?

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)

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\}

Think of rotations on a sphere

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

My quantum algorithms use such unitary matrices - they are implementing Riemannian gradient descent

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}

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

| \psi\rangle \mapsto \ket{\psi'}= 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?

Text

We need to find a tangential direction which lowers the energy of $\ket\psi$

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

| \psi\rangle \mapsto \ket{\psi'}= e^{is A}|\psi\rangle

Let's see how a direction $A=A^\dagger$ changes the energy of $\ket\psi$

Tr(ABC) = Tr(CAB)\\ \Downarrow \\ Tr(|\psi\rangle\langle \psi| [H,A] ) = Tr([|\psi\rangle\langle \psi|,H]A)

\partial_s \langle \psi| e^{-isA}He^{isA}| \psi\rangle_{|s=0} = -i\langle \psi| [A,H]| \psi\rangle

=- iTr\left[|\psi\rangle\langle\psi| [A,H]\right]

This bracket is called the Riemannian gradient

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.

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:

Define $|{\psi_k}\rangle = U_k |0\rangle$
Use $e^{is |\psi_k\rangle\langle\psi_k|} = U_ke^{is |0\rangle\langle0|}U_k^\dagger $
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)

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:

Define $|{\psi_k}\rangle = U_k |0\rangle$
Use $e^{is |\psi_k\rangle\langle\psi_k|} = U_ke^{is |0\rangle\langle0|}U_k^\dagger $
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:

Define $|{\psi_k}\rangle = U_k |0\rangle$
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)

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

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

Solve the unitary synthesis problem in all these cases through Riemannian gradients!

Double-bracket quantum algorithms

Coherently implement Riemannian gradient steps
Give rigorous unitary synthesis for
- imaginary-time evolution
- quantum signal processing
- diagonalization unitaries
- Grover's as an approximation to imaginary-time evolution
Training quantum circuits from data doesn't work well, unlike in classical machine learning applications. However those variational learning methods are great for warm-starting double-bracket quantum algorithms!

Tell me when not fast enough? Get stuck? Something else?

Your input is needed to improve them!

N. Ng

Z. Holmes

R. Zander

R. Seidel

Y. Suzuki

B. Tiang

J. Son

S. Carrazza

Stay in touch on LinkedIn:

Text

Marek Gluza

The double-bracket quantum algorithms roadmap

I grew up around these mountains where Poland meets Czech Republic and Slovakia (in Europe)

June '22: Single-author double-bracket proposal

This talk is an overview of 4 years of my research on double-bracket quantum algorithms:

Never worked on quantum algorithms before $\rightarrow$ 9 papers, 1 experiment
Never participated in a research program $\rightarrow$ lead a collaboration of 25 co-authors
It was the mathematical observations guiding us $\rightarrow$ Riemannian geometry is key!

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