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.

0. Problem choice:
What challenge to take up?

Materials science?

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

Why materials science? Imagine improving photovoltaics by 1%.

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

How? Quantum circuit synthesis.

0. Problem choice:
What challenge to take up?

Materials science?

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

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!

[1]

[9]

[8]

[5]

[2]

[3,4]

[6]

[7]

Click these links at slides.com/marekgluza

October '21: Arrived to Singapore

How to go about designing quantum algorithms?

Now: Not many distinct quantum algorithms
https://quantumalgorithmzoo.org/
Why?

Classically: Reduction to decision problems
Quantumly: Is $\langle\psi|H|\psi\rangle$ lower than ...?
Optimization: Only quadratic speed-up?

"Grand unification of quantum algorithms" https://arxiv.org/abs/2105.02859 the block-encoding scheme for accessing $H$
This talk: Another unification based on Riemannian optimization and using geodesics $e^{isH}$ to access $H$
Double-bracket quantum algorithms give intuition why unification possible: $\langle\psi|H|\psi\rangle$ is optimization so interpret through Riemannian gradients

Click links at slides.com/marekgluza

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

="\overline{v}^\top H v"

Marek Gluza

The double-bracket quantum algorithms roadmap

...the outline of the talk

Today, I will tell you about double-bracket quantum algorithms

Part 1:

What are quantum algorithms?
What is quantum computing?

Part 3:

What can they be used for?
What are good tasks for quantum computers?

Part 2:

Riemannian geometry of the unitary group

Part 4: Just basic properties of the unitary group facilitate quantum algorithm design

New quantum algorithms based on Riemannian optimization

Marek Gluza

NTU Singapore

Scan QR code or go to slides.com/marekgluza

Watch on youtube: https://www.youtube.com/watch?v=PLVkuqPemVs

What is a quantum computer?

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

J. Leonard (TU Vienna) pointing where neutral atoms will be trapped

R. Dumke and S. Carrazza discussiong the CQT lab

Some investors speculate their money on it - RGTI, IONQ...

We can make it!

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 $|v\rangle \in \mathcal V$

We take more qubits

Apply patterned EM radiation
Reproducible: Use FPGA

Reversible: Physics challenge; keep cold

We can make it!

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

Physics papers from the 90's

For good or for worse quantum computing is influenced by physics...

How to go about designing quantum algorithms?

Now: Not many distinct quantum algorithms
https://quantumalgorithmzoo.org/
Why?

00's Shor: Why have not more algorithms been found?
https://cs.brynmawr.edu/Courses/cs380/fall2012/Shor2003.pdf
Either because the easy ones are efficient classically or we have no good language

Classically: Reduction to decision problems
Quantumly: Is $\langle\psi|H|\psi\rangle$ lower than ...?
Optimization: Only quadratic speed-up?

Click links at slides.com/marekgluza

90's Shor: quantum computers can break cryptography

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

="\overline{v}^\top H v"

Why materials science? Imagine improving photovoltaics by 1%.

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

How? Quantum circuit synthesis.

This is worth being hyped!

Materials science?

What is a quantum algorithm?

What is a quantum computer?

It's an experimental setup A which includes a quantum system B and that setup A allows to manipulate the quantum state of B.

(This monumental structure is the insides of the dilution refrigerator shielding qubits from noise)

A - basket

B - fruits

Cat - noise

A - quantum computer

B - qubits

C - noise

Quantum Dynamic Programming in Qibo

How to use quantum computers programmed by quantum information?

Can quantum states program a quantum computer?

Can quantum information be the source code?
What would it be good for?

Introduction

What is a dynamic quantum algorithm?

Dynamic Quantum Algorithm

Consider a data qubit and an instruction qubit
Make an infinitesimal rotation by Heisenberg hamiltonian on these 2 qubits
Discard the instruction qubits and prepare a fresh copy of that instruction state
Repeat the Heisenberg evolution and fresh preparation
If the infinitesimal rotation duration is dt then after M steps we will have
The circuit will be exactly the same even if we use different qubit states

\sigma \to e^{idtM\rho}\sigma e^{-idtM\rho} + O(M^{-1})

XX + YY + ZZ

\sigma

\rho

Density Matrix Exponentiation

Consider a data qubit and an instruction qubit
Make an infinitesimal rotation by applying partial SWAP ($\delta$SWAP) gate, realized by Heisenberg interaction
Discard the instruction qubits and prepare a fresh copy of that instruction state.
Since density matrix $\rho$ is Hermitian, we can use it as an operator: $\sigma \to e^{-i\rho\delta}\sigma e^{i\rho\delta}$
Repeat the partial SWAP and fresh preparation
If the infinitesimal rotation duration is dt then after M steps we will have

\sigma \to e^{idtM\rho}\sigma e^{-idtM\rho} + O(M^{-1})

XX + YY + ZZ

\sigma

\rho

Density Matrix Exponentiation

Since density matrix $\rho$ is Hermitian, we can use it as an operator
DME implements am infinitesimal unitary rotation about an axis defined by instruction qubit 𝜌 on data qubit 𝜎:\[e^{-i\rho\delta}\sigma e^{i\rho\delta}\]
After Trotterization, DME implements: $U = e^{-i\rho\theta}$
Again, this is dynamic because $\rho$ is only determined at runtime

Dynamic Quantum Algorithm

Quantum algorithms that are dynamic: their operation is revealed on runtime.
Dynamic quantum algorithm: Algorithm that take an unknown density matrix and design a circuit that will meaningfully use this quantum information as an instruction
Kimmel et al. (arxiv 1608.00281) shows that this model is basis of a universal model for quantum computation
Example usage: there is a universal circuit that compute the Schmidt spectrum

Static vs Dynamic

Kjaergaard et al., 2001.08838

Usual quantum computing is static: To change operation, we have to change the circuit
Our approach (dynamic quantum computing) is dynamic: To change operation, only need to change instruction qubit

Existing literature

Kjaergaard et al., 2001.08838

Motivation for

Quantum Dynamic Programming

What is Quantum Dynamic Programming (QDP) good for?

What is a quantum recursion

Definition:
More complicated, dynamic case:

As we are taking more steps, the recursive state is instructing the propagation
Problem:
- What is a meaningful $U(\psi)$?
- To calculate $\ket {\psi''}$ will we need to recalculate $\ket {\psi'}$ and calculate $U(\psi')$?
- For higher order terms, do we need to recalculate all lower terms?

\ket{\psi'} = U(\psi) \ket \psi\\

U(\psi) \neq U(\psi')\\ \text{if } \psi \neq \psi'

\ket{\psi''} = U(\psi') \ket{\psi'}\\

Quantum recursion examples

Solution: QDP shows how to do it in a quantum coherent way.
Examples of quantum recursions:
- Grover rotation: \[G=id - 2\ket\psi\bra\psi\]
  $G(ψ1)=U0†G(ψ0)U0=id−2∣ψ1⟩⟨ψ1∣G^{(\psi_1)} = U_0^\dagger G^{(\psi_0)}U_0=\mathbb{id} - 2\ket{\psi_1}\bra{\psi_1}$
- Grover search: Add static unitaries Q and R that don’t depend on the state:
  $U^{(\psi)} = QG^{(\psi)}R$ with $Q,R\in U(D)$
- Density Matrix Exponentiation: Explain dynamic Grover rotation implementation: \[G = e^{i\psi s}\]

Quantum recursion examples

Reduced group commutator: Approximate commutator evolution and similar structures that evolve with density matrix.
\[e^{isD}e^{it\rho}e^{-isD}\]
We can use this to diagonalize
$eisDeitρe−isDe^{isD}e^{it\rho}e^{-isD}$

Marek DBI paper, arxiv 2206.11772

QDP paper, arxiv 2403.09187

Unfolding quantum recursions

Consider Grover reflector: $G=id - 2\ket\psi\bra\psi$

$Q,R \in U(D)$, initial state $\ket{\psi_0}$, $\ket{\psi_{k+1}} = QG^{(\psi_k)}R\ket{\psi_k}$

To get $\ket{\psi_1}$ is simple:

\begin{align*} G^{(\psi_1)} &= U_0^\dagger G^{(\psi_0)}U_0\\ &=\mathbb{1} - 2\ket{\psi_1}\bra{\psi_1}\\ &=\mathbb{1} - 2U_0 \ket{\psi_0}\bra{\psi_0}U_0^\dagger\\ &=U_0(\mathbb{1}-2\ket{\psi_0}\bra{\psi_0})U_0^\dagger\\ \ket{\psi_2} &= QG^{(\psi_1)}R\ket{\psi_1}\\ &=\boxed{QU_0(\mathbb{1}-2\ket{\psi_0}\bra{\psi_0})U_0^\dagger R QG^{(\psi_0)}R\ket{\psi_0}} \end{align*}

\begin{align*} U(\psi_0) &= QG^{(\psi_0)}R\\ \ket{\psi_1} &= QG^{(\psi_0)}R\ket{\psi_0} \end{align*}

However, it's more complicated to get $\ket{\psi_2}$. Ordinary idea:

Motivation for QDP

Unfolding the recursion make the depth grow exponentially
Using placeholder memory is not straightforward with quantum computing
Notice that the unitary $U(\psi)$ depends on $\ket\psi$. Can we use the state to steer the evolution?
Quantum Dynamic Programming (QDP), a framework that uses copies of the recursive state to implement the recursion step unitary
This is dynamic because the instruction state is revealed only on runtime
QDP yields exponential reduction in circuit depth than when unfolding

Quantum Dynamic Programming

J. Son et al, arxiv 2403.09187

Quantum Dynamic Programming

Simple case: Recursion step unitary that make a single memory call
\[ U^{(\mathcal{N},\rho)} = V_2e^{i\mathcal{N}(\rho)}V_1\] where $\mathcal{N}$ is the operator of the memory call
General case: Repeat this many times for the same instruction state
Memory call: the step is asking for memory (instruction state $\rho$)
Problem: We can't implement this naturally in qunatum mechanics

Quantum Dynamic Programming

Solution: QDP approximate this memory call unitary:
QDP does this by using memory usage query:

where $N$ is the unitary of the memory usage query
Memory usage query: consume (trace out) an instruction state
Repeat this procedure M times, we obtain

\mathcal{E}_s^{(N,\rho)}(\sigma) = Tr_\rho [e^{-iNs} (\rho\otimes\sigma) e^{iNs}]

E^{(\mathcal{N},\rho)}(\sigma) = e^{i\mathcal{N}(\rho)} \sigma e^{-i\mathcal{N}(\rho)}

\mathcal{E}^{(\mathcal{N},\rho,M)}_{QDP} \coloneqq (\mathcal{E}^{(\mathcal{N},\rho)}_{1/M})^M = E^{(\mathcal{N},\rho)} + O(1/M)

Density Matrix Exponentiation (DME) in Qibo

How does DME model work in Qibo?

Kjaergaard et al., arxiv 2001.08838

Memory usage query in DME

Goal: DME approximate the rotation (the memory call):
\[U = e^{-i\rho\theta}\]
DME replace the infinitesimal angle memory call unitary (where $\delta = \theta/N$): \[\sigma\to e^{-i\rho\delta}\sigma e^{i\rho\delta}\] by the memory usage query \[\sigma\to\text{Tr}_\rho[e^{-i\text{SWAP}\delta}(\sigma\otimes\rho) e^{i\text{SWAP}\delta}]\]
DME then does $N$ memory usage queries (Trotterization):

DME(\rho,N,\theta) = e^{-i\rho\theta} + O\left(\frac{\theta^2}{N}\right)

DME flowchart

\text{Tr}_\rho[e^{-i\text{SWAP}\delta}(\sigma\otimes\rho) e^{i\text{SWAP}\delta}] \\ = e^{-i\rho\delta}\sigma e^{i\rho\delta}+O(\delta^2)

Problem of Entanglement

To use quantum info from $\rho$, we need it to interact and correlate with $\sigma$
How do we avoid entanglement between $\rho$ and $\sigma$?
Example: the Bell state circuit create entanglement

How DME avoid entanglement

DME uses a series of partial SWAP gate, realized by Heisenberg interaction: $H_{\text{Heisenberg}} = e^{it{(XX + YY + ZZ)}}$
DME relies on the relation:
Where, after the partial SWAP, by tracing out the instruction qubit, we get in first order the desired rotation
The data qubit is in a pure state since the rotation is infinitesimal

DME in Qibo

models/qdp/quantum_dynamic_programming.py

DME tutorial notebook

QDP code structure

Github PR #1302

QDP code structure

There are 2 important parts of a QDP circuit:
- Memory call circuit: How many instruction qubit to use, how to get N copies of the instruction qubit
- Memory usage query (circuit): Define the operator used to generate the unitary rotation (in the case of DME the 𝛿SWAP)

Conclusion

How to use quantum computers programmed by quantum information?
Can quantum states program a quantum computer? Can quantum information be the source code?
What would it be good for?
What is a dynamic quantum algorithm? What is QDP?
What is DME? How does DME work?
The QDP code structure?