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

The double-bracket quantum algorithms journey

June '22: Single-author double-bracket proposal

Growth achieved during 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]

October '21: Arrived to Singapore

Game changer - NTU's PPF 200k grant

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

Next: What is a quantum computer

This is not a quantum computer

but just its casing

Later: Why this visualizes the operations on a quantum computer

(Riemannian geometry)

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 $v = |v\rangle \in \mathcal V$

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?

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...

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

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

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

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

Next: Why a watermelon?

This visualizes the operations on a quantum computer

(Riemannian geometry)

Riemannian geometry is a generalization of the geometry of the sphere
For 1 qubit the sphere is a great model of quantum operations
In higher dimensions many parallels remain e.g. set of quantum operations is compact (the watermelon fits into the hand) but many get lost e.g. non-convex submanifolds (green colors on the watermelon)
If watermelon is placed on the table then the table is tangent to it

This visualizes the operations on a quantum computer

(Riemannian geometry)

Riemannian geometry is a generalization of the geometry of the sphere
For 1 qubit the sphere is a great model of quantum operations
In higher dimensions many parallels remain e.g. set of quantum operations is compact (the watermelon fits into the hand) but many get lost e.g. non-convex submanifolds (green colors on the watermelon)
If watermelon is placed on the table then the table is tangent to it

Next: Let's see how to use geometry to design 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

SU(d) = \{ M\in \mathbb C^{d\times d}: M^{-1} = M^\dagger, \mathrm{det}(M)=1\}

M \in SU(d) \Rightarrow \exist H=H^\dagger: M = e^{is H}

Fact 2. Any quantum gate arises as a point on a geodesic, the 'velocity' is the physical interaction $H$ of qubits

Think of rotations on a sphere

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}

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\|}

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

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

Examples

Trade bonding by HSBC
Random sampling
Mixed-integer programming? Shortest-vector problem? Etc.

When a lot of people can gain by having something to say, some will not resist the temptation to speak without having much to tell which creates hype

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?

Riemannian 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

(Riemannian optimization)

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

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

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

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 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

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\|}

2. Unitary synthesis:
How to do it?

3. Circuit compilation:
What gates to do?

0. Problem choice:
What challenge to take up?

We need an optimization solver, i.e. we need to find ways to lower $\bra\psi H\ket \psi$

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\|}

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 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?

Game changer - NTU's PPF 200k grant came in: Travel and hire

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\|}

2. Unitary synthesis:
How to do it?

3. Circuit compilation:
What gates to do?

0. Problem choice:
What challenge to take up?

We need an optimization solver, i.e. we need to find ways to lower $\bra\psi H\ket \psi$

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\|}

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\|}

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

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

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!

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"

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

October '21: Arrived to Singapore

...a few reflections

I grew up below this mountain where Poland meets Czech Republic and Slovakia (in Europe)

June '22: Single-author double-bracket proposal

Why so slow?

Need help

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 gometry is key!

Marek Gluza

The double-bracket quantum algorithms roadmap

October '21: Arrived to Singapore

...a few reflections

I grew up below this mountain 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 gometry is key!

Text

Why so slow?

Need help

[1]

[9]

[8]

[5]

[2,3]

[4]

[6]

[7]

Z. Holmes

Y. Suzuki

B. Tiang

J. Son

N. Ng

R. Zander

R. Seidel

S. Carrazza

Marek Gluza

The double-bracket quantum algorithms roadmap

October '21: Arrived to Singapore

...a few reflections

I grew up below this mountain 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 gometry is key!

Text

Why so slow?

Need help

[1]

[9]

[8]

[5]

[2,3]

[4]

[6]

[7]

Z. Holmes

Y. Suzuki

B. Tiang

J. Son

N. Ng

R. Zander

R. Seidel

S. Carrazza

Marek Gluza

The double-bracket quantum algorithms roadmap

October '21: Arrived to Singapore

June '22: Single-author proposal

April '22: Dynamic programming?

April 2023: Yes!

April 2024: Arxiv

Why so slow?

R. Takagi

J. Son

N. Ng

How many quantum gates are needed?

Game changer - NTU's PPF 200k grant came in: Travel and hire

S. Carrazza

August '24: Very few!

August '23: Maybe few?

...a few reflections

Need help

Marek Gluza

The double-bracket quantum algorithms roadmap

October '21: Arrived to Singapore

January '22: First outside discussion of DBQA

5 am with D. Gosset from Toronto lead to most important paper in December 2024

June '22: Single-author proposal

April '22: Dynamic programming?

April 2023: Yes!

April 2024: Arxiv

Lessons from stage 1

When developing something outside the toolbox:

The target audience are experts - they will be your collaborators for the follow-ups
We each are busy with our own ideas "Oh, I saw your paper on the arxiv but then I forgot it."

Why so slow?

R. Takagi

J. Son

N. Ng

How many quantum gates are needed?

Game changer - NTU's PPF 200k grant came in: Travel and hire

S. Carrazza

August '24: Very few!

August '23: Maybe few?

...a few reflections

Marek Gluza

The double-bracket quantum algorithms roadmap

October '22: Arrived to Singapore

January '22: First outside discussion of DBQA

5 am with D. Gosset from Toronto lead to most important paper in December 2024

June '22: Single-author proposal

April '22: Dynamic programming?

April 2023: Yes!

April 2024: Arxiv

Why so slow?

R. Takagi

J. Son

N. Ng

How many quantum gates are needed?

Game changer - NTU's PPF 200k grant came in: Travel and hire

S. Carrazza

August '24: Very few!

August '23: Maybe few?

Lessons from stage 2

Developing it further lead to

Chance for new people to have impact
Surprising reviewer bias against progress "There were no important algorithms in the last 10 years so this one must be unimportant too."

rollercoaster

Z. Holmes

Y. Suzuki

B. Tiang

J. Son

N. Ng

December '24: DB-QITE - our most important paper so far

R. Zander

R. Seidel

...a few reflections

https://arxiv.org/abs/2412.04554

Marek Gluza

The double-bracket quantum algorithms roadmap

October '22: Arrived to Singapore

January '22: First outside discussion of DBQA

5 am with D. Gosset from Toronto lead to most important paper in December 2024

June '22: Single-author proposal

April '22: Dynamic programming?

April 2023: Yes!

April 2024: Arxiv

Why so slow?

R. Takagi

J. Son

N. Ng

How many quantum gates are needed?

Game changer - NTU's PPF 200k grant came in: Travel and hire

S. Carrazza

August '24: Very few!

August '23: Maybe few?

Lessons entering stage 3

Riemannian optimization is a unifying principle of quantum algorithms!

rollercoaster

Z. Holmes

Y. Suzuki

B. Tiang

J. Son

N. Ng

December '24: DB-QITE - our most important paper so far

impact?

R. Zander

R. Seidel

...a few reflections

arxiv:2504.01077

arxiv:2507.15065

Marek Gluza

The double-bracket quantum algorithms roadmap

October '22: Arrived to Singapore

January '22: First outside discussion of DBQA

5 am with D. Gosset from Toronto lead to most important paper in December 2024

June '22: Single-author proposal

April '22: Dynamic programming?

April 2023: Yes!

April 2024: Arxiv

Why so slow?

R. Takagi

J. Son

N. Ng

How many quantum gates are needed?

Game changer - NTU's PPF 200k grant came in: Travel and hire

S. Carrazza

August '24: Very few!

August '23: Maybe few?

Lessons entering stage 3

Riemannian optimization is a unifying principle of quantum algorithms!

rollercoaster

Z. Holmes

Y. Suzuki

B. Tiang

J. Son

N. Ng

December '24: DB-QITE - our most important paper so far

impact?

R. Zander

R. Seidel

...a few reflections

arxiv:2504.01077

arxiv:2507.15065

C. Mostajeran

Riemannian optimization well established in classical numerical analysis:

Use Riemannian conjugate gradients to formulate new quantum algorithms
How to use a Riemannian Hessian?

Lessons entering stage 3

Riemannian optimization is a unifying principle of quantum algorithms!
Even more people have impact

Marek Gluza

The double-bracket quantum algorithms roadmap

October '22: Arrived to Singapore

January '22: First outside discussion of DBQA

5 am with D. Gosset from Toronto lead to most important paper in December 2024

June '22: Single-author proposal

April '22: Dynamic programming?

April 2023: Yes!

April 2024: Arxiv

Why so slow?

R. Takagi

J. Son

N. Ng

How many quantum gates are needed?

Game changer - NTU's PPF 200k grant came in: Travel and hire

S. Carrazza

August '24: Very few!

August '23: Maybe few?

Lessons entering stage 3

Riemannian optimization is a unifying principle of quantum algorithms!
Even more people have impact

rollercoaster

Z. Holmes

Y. Suzuki

B. Tiang

J. Son

N. Ng

December '24: DB-QITE - our most important paper so far

C. Mostajeran

impact?

R. Zander

R. Seidel

...a few reflections

Marek Gluza

October '22: Arrived to Singapore

January '22: First outside discussion of DBQA

5 am with D. Gosset from Toronto lead to most important paper in December 2024

June '22: Single-author proposal

April '22: Dynamic programming?

April 2023: Yes!

April 2024: Arxiv

Why so slow?

R. Takagi

J. Son

N. Ng

S. Carrazza

August '24: Very few!

August '23: Maybe few?

rollercoaster

Z. Holmes

Y. Suzuki

B. Tiang

J. Son

N. Ng

December '24: DB-QITE - our most important paper so far

impact?

R. Zander

R. Seidel

...a few reflections

Riemannian optimization well established in classical numerical analysis:

Use Riemannian conjugate gradients to formulate new quantum algorithms
How to use a Riemannian Hessian?

Materials science?

Braket notation

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

1 qubit

2 qubits

*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

Braket notation

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

1 qubit

2 qubits

3 qubits

4 qubits

And you get the idea lah

\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

*Dirac was one of the smartest quantum physicist in history

= a e_1\otimes e_1\otimes e_1 + b e_2\otimes e_1\otimes e_1+\ldots

$\leftarrow$ quickly unreadable!

Mathematics in quantum computing

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

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 $v = |v\rangle \in \mathcal V$

Based on approximation derived by physicists to an infinite Hilbert space

To preserve normalization of probability: These operations are unitary matrices

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

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

We take more qubits

We can make it!

Braket notation $v = |v\rangle \in \mathcal V$

Scalar products, projections etc.

Braket notation

*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 = \begin{pmatrix} 1\\0\end{pmatrix}, | 1\rangle :=e_2= \begin{pmatrix} 0\\1\end{pmatrix}

Why braket? Probabilities of quantum events are overlaps of vectors

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

\langle e_1,Ae_2\rangle =\langle Ae_1,e_2\rangle = \langle 0 |A|1\rangle = 0.15+2i

Expectation values of observations are related to Hermitian matrices $ A = A^\dagger $

\langle\psi|A|\psi\rangle

(measurement of a quantum system in $|0\rangle$ will never point to it being in $|1\rangle$)

(quantum superpositions can feature complex numbers)

Braket notation is useful

*Dirac was one of the best physicists in history

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

| 0 0\rangle :=e_1\otimes e_1, | 10\rangle :=e_2\otimes e_1

P_{1,0} = \langle\psi|A|\psi\rangle = \langle\psi| (e_2\otimes e_1) \cdot (\overline{e_2\otimes e_1})^\top|\psi\rangle

Example: The probability to find the quantum computer in state $ |10\rangle$ is the mean of the matrix $A=|10\rangle\langle 10| = (e_2\otimes e_1) \cdot (\overline{e_2\otimes e_1})^\top$

P_{1,0} = \langle\psi|10\rangle\langle10|\psi\rangle = a\overline{b} \langle10|10\rangle\langle 10|00\rangle+|b|^2 \langle 10|10\rangle^2 +\ldots

P_{1,0} = \langle\psi|10\rangle\langle10|\psi\rangle = |b|^2

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?

Double-bracket quantum algorithms:

Systematic framework for unitary synthesis

|\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)

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

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

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

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?

Double-bracket quantum algorithms:

Systematic framework for unitary synthesis

|\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)

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

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

Product formula approximation:

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

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?

|\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?

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

Two theorems 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:

1. Mean energy $E_k := \langle \psi_k| H|{\psi_k}\rangle$ decreases as

$E_{k+1} = E_k - 2s V_k + O(s^2)$

where $V_k := \langle \psi_k| H^2|{\psi_k}\rangle - E_k^2$

2. Ground-state fidelity $F_k := |\langle \lambda_0| {\psi_k}\rangle|^2$ increases as

$F_{k} \ge 1 - q^k$

where $q = 1 - O( \Delta F_0 / \|H\|^3)$

https://arxiv.org/abs/2412.04554

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

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

Goals?

Realistic?

Intuition?

Quantum computer

Quantum algorithm

Cooling

Natural

0

C

[1] Double-bracket quantum algorithms for diagonalization

YES!!!!

Singapore's

quantum computer

R. Dumke

S. Carrazza

Optimizing a random guess to cool gets stuck

Can't do material science?! :(

DBQA moves it ahead!

[3]

We know how to make it run fast!

Quantum gates

N. Ng

Exactly how good is it?

[2]

Double-bracket quantum algorithms

coherently implement Riemannian gradient steps
give rigorous unitary synthesis for
- imaginary-time evolution
- quantum signal processing
- diagonalization unitaries
Grover's algorithm is an approximation to imaginary-time evolution

J. Son

B. Tiang

S. Carrazza

R. Seidel

R. Zander

Z. Holmes

Y. Suzuki

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

Your input is needed to improve them!

N. Ng

What about other methods?

0

Universal gate set:

single qubit rotations + generic 2 qubit gate

Universal gate set can approximate any unitary

What is a universal quantum computer?

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

quantum compiling approximates unitaries with circuits

\Rightarrow U \approx U_c

Quantum compiling

2x2 unitary matrix - use Euler angles

4x4 unitary matrix - use KAK decomposition + 3x CNOT formula

\text{CNOT} = |0\rangle\langle 0|\otimes \begin{pmatrix} 1 & 0\\ 0 &1\end{pmatrix} + |1\rangle\langle1 |\otimes \begin{pmatrix} 0 & 1\\ 1 &0\end{pmatrix}

https://arxiv.org/abs/quant-ph/0308006v3

=

2 qubit unitary

Canonical

KAK decomposition, Brockett's work etc

=

2 qubit unitaries modulo single qubit unitaries are a 3 dimensional torus

Quantum compiling

$2$ qubits - $4\times 4$ unitary matrix - use KAK decomposition + $3$ CNOT formula

Quantum compiling

$1$ qubit - $2\times 2$ unitary matrix - use Euler angles

$n$ qubits - $2^n$ unitary matrix - use quantum Shannon decomposition + $O(4^n)$ CNOT formula

Variational quantum eigensolver

https://arxiv.org/abs/quant-ph/0308006v3

0

+

This works but is inefficient

This is efficient but doesn't work

Open: fill this gap!

\hat W_k = [\hat D_{k-1},\hat H_{k-1}]

\hat H_k = e^{s_k \hat W_k} \hat H_{k-1} e^{-s_k \hat W_k}

Double-bracket iteration

s_k

Rotation durations:

Input:

\hat H_0

\hat D_k

Diagonal generators:

A new approach to diagonalization on a quantum computer

Great: we can diagonalize

How to quantum compile?

New quantum algorithms based on Riemannian optimization

Scan QR code or go to slides.com/marekgluza

The double-bracket quantum algorithms roadmap

Click these links at slides.com/marekgluza

The double-bracket quantum algorithms journey

The double-bracket quantum algorithms roadmap

Next: What is a quantum computer

This is not a quantum computer

Later: Why this visualizes the operations on a quantum computer

What is a quantum computer?

What is a quantum computer?

What is a quantum computer?

What is a quantum algorithm?

What is a quantum computer?

Braket notation

What is a universal quantum computer?

It is a machine for realizing such linear combinations in nature

The double-bracket quantum algorithms roadmap

Next: Why a watermelon?

This visualizes the operations on a quantum computer

This visualizes the operations on a quantum computer

Next: Let's see how to use geometry to design quantum algorithms

The double-bracket quantum algorithms roadmap

Materials science?

Click links at slides.com/marekgluza

Materials science?

Riemannian 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

The double-bracket quantum algorithms roadmap

Double-bracket flows

Why double a bracket?

The double-bracket quantum algorithms roadmap

Design choices for optimization

Energy filtering in quantum optimization

Design choices for optimization

Double-bracket quantum algorithms

Click these links at slides.com/marekgluza

Double-bracket quantum algorithms

Click these links at slides.com/marekgluza

Click links at slides.com/marekgluza

Double-bracket quantum algorithms

The double-bracket quantum algorithms roadmap

The double-bracket quantum algorithms roadmap

The double-bracket quantum algorithms roadmap

The double-bracket quantum algorithms roadmap

The double-bracket quantum algorithms roadmap

The double-bracket quantum algorithms roadmap

rollercoaster

The double-bracket quantum algorithms roadmap

rollercoaster

impact?

The double-bracket quantum algorithms roadmap

rollercoaster

impact?

The double-bracket quantum algorithms roadmap

rollercoaster

impact?

rollercoaster

impact?

Materials science?

Braket notation

Braket notation

Mathematics in quantum computing

Braket notation

Braket notation is useful

Double-bracket quantum algorithms

What about other methods?

Double-bracket iteration