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

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]

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[2]

[3,4]

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Click these links at slides.com/marekgluza

October '21: Arrived to Singapore

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

What is a quantum algorithm?

0

C

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

(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

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

0

C

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

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

Guidelines for using quantum computing

Stage 1: Think. What is the goal?

Important problems that are difficult yet doable.

Encode what we know in $\vec{v}_{input}$.

Decode information from $\vec{v}_{output}$.

Find effective heuristics to reduce the runtime of rotations $R_k$

Find rotations such that

$\vec{v}_{output} \approx R_1 R_2 \ldots R_n \vec{v}_{input}$

Stage 2: Design. How to encode task in quantum mechanics?

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

Stage 4: Run it.
What instructions to send?

Stage 3: Algorithm. How to find $\vec{v}_{output}$?

Guidelines for extracting utility from quantum computing

Stage 1: Think. What is the goal?

Bridge between needs of humans and technological feasibility.
Exchange domain expertise with industry partners.
My work: General-purpose optimization solver in quantum computing based on non-Euclidean geometry.
My expertise: Algorithms optimized for execution on leading prototypes.

Stage 4: Run it.
What instructions to send?

Stage 2: Design. How to encode task in quantum mechanics?

Stage 3: Algorithm. How to find $\vec{v}_{output}$?

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

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

Observation leading to double-bracket quantum algorithms: Earth is not flat. I.e., 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.

Stage 3: Algorithm. How to find $\vec{v}_{output}$?

Riemannian geometry underlying quantum algorithms

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

Stage 3: Algorithm. How to find $\vec{v}_{output}$?

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

Stage 3: Algorithm. How to find $\vec{v}_{output}$?

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)

Stage 3: Algorithm. How to find $\vec{v}_{output}$?

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

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}

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

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

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.

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

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

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?

g

W

iH

iA

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

W

iH

iA

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}

Double-bracket flows

\partial_t O(t)= i[H,O(t)]

\partial_t \rho(t)= i[H,\rho(t)]

\partial_t A(t)= [A(t),[A(t),B]]

H \sim -i[A(t),B]

Heisenberg equation	Linear, variable: observable
Schroedinger equation	Linear, variable: density matrix
Double-bracket flow	Non-linear, variable: density matrix or observable The solution is a unitary rotation because

Why double a bracket?

\partial_t \hat A(t) = i [ \hat H(t), \hat A(t)]

\partial_t \hat A(t) = [ \hat H(t), [\hat H(t), \hat A(t)]]

=

2 qubit unitary

Canonical

\partial_t \hat A(t) = [ \hat A(t), [\hat A(t), \hat B]]

Double-bracket quantum algorithms

are inspired by double-bracket flows

and allow to perform optimization through short quantum computations

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

\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}

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

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\|}

0. Problem choice:
What challenge to take up?

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

3. Circuit compilation:
What gates to do?

2. Unitary synthesis:
How to do it?

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)

A crisis in quantum algorithm design:

Will quantum computers be fast?

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?

Adiabatic "QAOA" approach:
Slowly change the Hamiltonian to "quantumly" tunnel into the solution - not fast enough see e.g. arxiv:2510.06337

See arxiv:2510.06337

Journalist questions vs. hype:

What was done?
Where are its limitations?
Why is it faster?

(Pros and cons of heuristics)

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?

Double-bracket quantum algorithms are guaranteed to converge. They can pick-up after learning of an optimized circuit has gotten stuck.

Brute-force idea:
Select geodesic directions and learn how long step lengths - will get stuck arxiv:1803.11173

See Nature Comm. referee reports of arxiv:1803.11173

A crisis in quantum algorithm design:

Will quantum computers be fast?

S. Carrazza

August '24: Very few!

August '23: Maybe few?

How many quantum gates are needed?

[3] arxiv:2408.03987

(Pros and cons of heuristics)

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

What is quantum signal processing?

|\psi'\rangle = e^{-\tau H} |\psi\rangle = \sum_k \langle\lambda_k|\psi\rangle e^{-\tau \lambda_k }|\lambda_k\rangle \rightarrow 0

|\psi'\rangle = \frac{e^{-\tau H} |\psi\rangle}{\|e^{-\tau H} |\psi\rangle\|} \rightarrow |\lambda_0\rangle

|\psi'\rangle = \frac{P(H) |\psi\rangle}{\|P(H) |\psi\rangle\|} \rightarrow |\lambda_0\rangle

P(x)=\prod_{k=1}^K(x-x_k)\approx e^{-\tau x}

|\psi'\rangle = \frac{P(H) |\psi\rangle}{\|P(H) |\psi\rangle\|} \rightarrow \frac{H^{-1}|\psi\rangle}{\|H^{-1} |\psi\rangle\|}

P(x)\approx 1/x

Example 1: Exponential function

Example 2: Polynomial approximation to exponential

Example 3: Polynomial approximation to inversion

P(H)=\prod_{k=1}^K(H-x_k)\approx e^{-\tau H}

Quantum signal processing maps between states according to a polynomial filter

|\psi'\rangle = \frac{P(H) |\psi\rangle}{\|P(H) |\psi\rangle\|}

Approach 2. Exponentials of commutators

Recursive ~ exponentially long circuits
Double-bracket quantum algorithms link to Riemannian gradients

Approach 1. Block-encodings

Probablistic ~ exponentially unlikely postselection
Grand unification of quantum algorithms https://arxiv.org/abs/2105.02859

|\psi'\rangle = \frac{P(H) |\psi\rangle}{\|P(H) |\psi\rangle\|} = U_\psi(P) |\psi\rangle

U_\psi(P)

we will consider $1-\tau H \approx e^{-\tau H}$
we will find $U_\psi(P = 1-\tau H)$
discuss how to generalize to any QSP

|\psi'\rangle = \frac{P(H) |\psi\rangle}{\|P(H) |\psi\rangle\|} = U_\psi(P) |\psi\rangle

U_\psi(P)

|\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 = (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

= - 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

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, let's see more next.

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

([|\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, let's see more next.

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

Observation. If there is a complex root $P(H) = H-z$ can be done by

$U_\psi(P) = e^{i\theta |\psi\rangle\langle\psi|}e^{s[|\psi\rangle\langle\psi|,H]}$

Result: Recursive iteration implements any QSP!

This starts looking like quantum signal processing, let's see more next.

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