Befriending the quantum computing disruption:

Lessons from double-bracket quantum algorithms

Marek Gluza

NTU Singapore

slides.com/marekgluza

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 is quantum computing?

Part 2: Riemannian geometry of the unitary group

Part 3: Optimization solvers from basic properties of the unitary group

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.

A - basket

B - fruits

Cat - noise

A - quantum computer

B - qubits

C - noise

What is a quantum computer?

Prodigious prospects of quantum computing

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
Easy ones are efficient classically? We have no good language?

Click links at slides.com/marekgluza

90's Shor: quantum computers can break cryptography
- You could make money opening pandora boxes
- How long can you keep opening them? (Post-quantum cryptography)

Useful and important problems are not obviously efficiently solvable on a quantum computer...

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.

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/

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?
Random sampling
Mixed-integer programming? Shortest-vector problem? Etc.

Classical computation

Difficult, important and doable problem

Millenials

What challenge to take up?

What is a quantum computer made of?

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.

Types of quantum computers:

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

Advantages:

A) Fabrication similar to Si pipelines
B) Naturally quantum
C) Large numbers
D) Leverage pecularities

It is hard to do anything useful, now, without knowing the low-level physics of the devices: Think vacuum tubes rather than CMOS

Disadvantages:

A) A-priori very noisy (but: arxiv:2408.13687)
B) Scaling-up (but: arxiv:2511.05465)
C) Unknown unknowns (arxiv:2412.15165)
D) More risky (but: arxiv:abs/2307.06617)

Types of quantum computers:

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

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.

What is a quantum computer made of?

It is hard to do anything useful, now, without knowing the low-level physics of the devices: Think vacuum tubes rather than CMOS

What is a good 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.

Outperforming your laptop:

A) Superconducting loops (Google)
B) Ionized atoms (Quantinuum)
C) Neutral atoms (Quera)

Initially:

What's the apriori quality of B?
How do you make B modular?
How good is setup A in manipulating B?
What does A leverage against noise?

What is a powerful 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.

Eventually:

How much error-correction required?
How many qubits can you cram?
Speed of elementary gates?
Overheads for real-world problems?

Utility-scale: When quantum computers will not be limited by noise in what they can do. We assume things will be just fine from then on! (?)

If you had a perfect quantum computer today - what would you use it for?

Outperforming your laptop:

A) Superconducting loops (Google)
B) Ionized atoms (Quantinuum)
C) Neutral atoms (Quera)

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?
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

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?
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

Why materials science? Imagine improving photovoltaics by 1%.

What is needed? Prepare low-energy state of the material and study its physics.

How? Optimization quantum algorithms and physics research.

This monumental structure is the insides of the dilution refrigerator, shielding qubits from noise and guiding control lines inside.

A - basket

B - fruits

Cat - noise

A - quantum computer

B - qubits

C - noise

Breaking RSA will be disruptive.
Materials science could be constructive.
Quantum computers efficiently simulate processes in nature and cooling of materials occurs in nature.
The entire community searched for general optimization speed-ups but rests hope on heuristics.

What will quantum computer do?

Symptomatic of detail complexity in quantum computing hardware: The 'casing' of the computer can be as important as the 'processor'.

Current phase: Break-even between wisdom and naivety.

https://www.forbes.com/sites/moorinsights/2020/04/30/googles-top-quantum-scientist-explains-in-detail-why-he-resigned/

A - quantum computer

B - qubits

C - noise

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.

What is the mathematical model?

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

Having $n$ qubits means tensor product structure $\mathcal V = (\mathbb C^2)^{\otimes n}$ and $d=2^n$

Quantum gates are unitary matrices

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

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

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 is quantum computing?

Part 2: Riemannian geometry of the unitary group

Part 3: Optimization solvers from basic properties of the unitary group

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^\dagger = -g

e^{s g}

g^\dagger = -g

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

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

Riemannian geometry is essential for quantum computation

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

e^{is A}e^{is B}e^{-is A}e^{-is B} \neq 1

e^{is A}

e^{isB}

e^{-isB}

e^{-is A}

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

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

Think of rotations on a sphere

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

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

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

U(d)

e^{ [A,B]}

$[A,B]^\dagger = -[A,B]$

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

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

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

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

Think of rotations on a sphere

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

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

e^{isA}e^{isB}e^{-isA}e^{-isB} = e^{-s^2[A,B]}+O(s^3)

e^{is A}

e^{isB}

e^{-isB}

e^{-is A}

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

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

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 is quantum computing?

Part 2: Riemannian geometry of the unitary group

Part 3: Optimization solvers from basic properties of the unitary group

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?

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?

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}

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 is quantum computing?

Part 2: Riemannian geometry of the unitary group

Part 3: Optimization solvers from basic properties of the unitary group

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

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

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!

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

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

Breather pause for time check. Next either:
A) 2 mins and 42 seconds to complete the process of unitary synthesis
B) or wrap up?

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

\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

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?

Product formula approximation:

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

= e^{i\sqrt{\tau}H}e^{i\sqrt{\tau}|\psi\rangle\langle\psi|}e^{i\sqrt{\tau}H}|\psi\rangle + 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?

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)

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)

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:

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