Foundations of quantum technologies · Girton College, Cambridge · 18^th September 2026

Optimal Hamiltonian control for variational quantum algorithms

on spin-qubit quantum processors

Christopher K. Long^1,2,3

¹University of Cambridge

²Hitachi Cambridge Laboratory

³Virginia Tech

Co-authors

Nicholas J. Mayhall³, Sophia E. Economou³, Edwin Barnes³, Crispin H. W. Barnes¹, Frederico Martins², David R. M. Arvidsson-Shukur², Normann Mertig²

VQAs […] appear to be the best hope for obtaining quantum advantage.

M. Cerezo et al., Nature Reviews Physics

“Variational quantum algorithms” (Sept. 2021), pp. 625–644. issn: 2522-5820. doi: 10.1038/s42254-021-00348-9

Variational quantum eigensolver Introduction & recap

Acceleration Hamiltonian-level algorithms

Robustness Leakage, dechorence, device imperfections

Scalability
Optimizability & Lie algebras

Contents04

Headline result

10⁵×

Faster & more noise robust

Improvement of our Hamiltonian-level approach over state-of-the-art gate-based approaches

Neglecting measurement time; measurement time is now the limiting factor for both speed and error rate.

Intro05

Section 01

Variational quantum eigensolver

Introduction & recap: what is a variational quantum eigensolver?

Variational quantum eigensolver

Parameter Space

Section 0107A

Variational quantum eigensolver

Parameter Space

|\psi(\theta)\rangle = U(\theta)|\psi_0\rangle

Hilbert Space

Section 0107B

Variational quantum eigensolver

Parameter Space

|\psi(\theta)\rangle = U(\theta)|\psi_0\rangle

Hilbert Space

E(\theta) = \langle\psi(\theta)|\hat{H}|\psi(\theta)\rangle

Energy Spectrum

Section 0107C

Variational quantum eigensolver

Quantum Processor

Classical Optimizer

Section 0108

Excessive runtimes

Single shot takes roughly 10–100 \mathrm{µs}

J. Tilly et al., Phys. Rep. 986, 1 (2022), doi:10.1016/j.physrep.2022.08.003Section 0109A

Excessive runtimes

Single shot takes roughly 10–100 \mathrm{µs}
2–20 hours per expectation value depending on accuracy you want

J. Tilly et al., Phys. Rep. 986, 1 (2022), doi:10.1016/j.physrep.2022.08.003Section 0109B

Excessive runtimes

Single shot takes roughly 10–100 \mathrm{µs}
2–20 hours per expectation value depending on accuracy you want
29–290 days per optimizer iteration

J. Tilly et al., Phys. Rep. 986, 1 (2022), doi:10.1016/j.physrep.2022.08.003Section 0109C

Excessive runtimes

Single shot takes roughly 10–100 \mathrm{µs}
2–20 hours per expectation value depending on accuracy you want
29–290 days per optimizer iteration
Many years for whole algorithm to run

J. Tilly et al., Phys. Rep. 986, 1 (2022), doi:10.1016/j.physrep.2022.08.003Section 0109D

Excessive runtimes

Single shot takes roughly 10–100 \mathrm{µs}
2–20 hours per expectation value depending on accuracy you want
29–290 days per optimizer iteration
Many years for whole algorithm to run
Parellelization is not feasible in the near term

J. Tilly et al., Phys. Rep. 986, 1 (2022), doi:10.1016/j.physrep.2022.08.003Section 0109E

Excessive runtimes

Single shot takes roughly 10–100 \mathrm{µs}
2–20 hours per expectation value depending on accuracy you want
29–290 days per optimizer iteration
Many years for whole algorithm to run
Parellelization is not feasible in the near term
Bring single shot time down to nanoseconds so iteration takes ~10 minutes

J. Tilly et al., Phys. Rep. 986, 1 (2022), doi:10.1016/j.physrep.2022.08.003Section 0109F

Decoherence

Amplitude damping

L=\sqrt{\frac{1}{T_1}}\,\sigma_-

Dephasing

L=\sqrt{\frac{1}{2T_2}}\,\sigma_z

Depolarizing

L_k=\frac{1}{2}\sqrt{p}\,\sigma_k,\ k\in\{x,y,z\}

Section 0110

Decoherence

C. K. Long et al., Phys. Rev. A 109, 042413 (2024), doi:10.1103/PhysRevA.109.042413
C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 2Section 0111A

Decoherence

C. K. Long et al., Phys. Rev. A 109, 042413 (2024), doi:10.1103/PhysRevA.109.042413
C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 2Section 0111B

Requirements

T_1\ge 1\,\text{s}

T_2\ge 100\,\text{ms}

p\le10^{-6}

to reach chemical accuracy 1.6\,\mathrm{mE_{h}}

Section 02

Acceleration

Hamiltonian-level algorithms: are they faster?

Device

placeholder

x z y

C. K. Long et al., npj Quantum Inf. 11, 113 (2025), doi:10.1038/s41534-025-01027-8
C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 3Section 0213A

Device

H(t) = -\frac{1}{2}\mu_B B_z \sum_{i=1}^{N} g_i \sigma_z^{(i)}

x z y

ZeemanStatic field B_z sets each qubit’s splitting.

C. K. Long et al., npj Quantum Inf. 11, 113 (2025), doi:10.1038/s41534-025-01027-8
C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 3Section 0213B

Device

H(t) = -\frac{1}{2}\mu_B B_z \sum_{i=1}^{N} g_i \sigma_z^{(i)} -\textcolor{#73407f}{\frac{1}{2}\mu_B B_x(t) \sum_{i=1}^{N} g_i \sigma_x^{(i)}}

x z y

ZeemanStatic field B_z sets each qubit’s splitting.

DriveAntenna delivers the transverse control field B_x(t).

C. K. Long et al., npj Quantum Inf. 11, 113 (2025), doi:10.1038/s41534-025-01027-8
C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 3Section 0213C

Device

H(t) = -\frac{1}{2}\mu_B B_z \sum_{i=1}^{N} g_i \sigma_z^{(i)} -\textcolor{#73407f}{\frac{1}{2}\mu_B B_x(t) \sum_{i=1}^{N} g_i \sigma_x^{(i)}} +\textcolor{#21918c}{\frac{1}{4}\sum_{i=1}^{N-1} J_i(t) \vec{\sigma}^{(i)} \cdot \vec{\sigma}^{(i+1)}}

x z y

ZeemanStatic field B_z sets each qubit’s splitting.

DriveAntenna delivers the transverse control field B_x(t).

ExchangeGate voltages tune nearest-neighbour coupling J_i(t).

C. K. Long et al., npj Quantum Inf. 11, 113 (2025), doi:10.1038/s41534-025-01027-8
C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 3Section 0213D

Device

H(t) = -\frac{1}{2}\mu_B B_z \sum_{i=1}^{N} g_i \sigma_z^{(i)} -\textcolor{#73407f}{\frac{1}{2}\mu_B B_x(t) \sum_{i=1}^{N} g_i \sigma_x^{(i)}} +\textcolor{#21918c}{\frac{1}{4}\sum_{i=1}^{N-1} J_i(t) \vec{\sigma}^{(i)} \cdot \vec{\sigma}^{(i+1)}}

ZeemanStatic field B_z sets each qubit’s splitting.

DriveAntenna delivers the transverse control field B_x(t).

ExchangeGate voltages tune nearest-neighbour coupling J_i(t).

C. K. Long et al., npj Quantum Inf. 11, 113 (2025), doi:10.1038/s41534-025-01027-8
C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 3Section 0213E

Gate-based computation

C. K. Long et al., npj Quantum Inf. 11, 113 (2025), doi:10.1038/s41534-025-01027-8
C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 3Section 0214

Hamiltonian-level computation

C. K. Long et al., npj Quantum Inf. 11, 113 (2025), doi:10.1038/s41534-025-01027-8
C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 3Section 0215

Molecule slide

C. K. Long et al., npj Quantum Inf. 11, 113 (2025), doi:10.1038/s41534-025-01027-8
C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 3Section 0216

Minimum evolution times

C. K. Long et al., npj Quantum Inf. 11, 113 (2025), doi:10.1038/s41534-025-01027-8
O. R. Meitei et al., npj Quantum Inf. 7, 155 (2021), doi:10.1038/s41534-021-00493-0
C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 3Section 0217

Virtual Z pulses

C. K. Long and C. H. W. Barnes, arXiv:2509.13453 (2025)
O. R. Meitei et al., npj Quantum Inf. 7, 155 (2021), doi:10.1038/s41534-021-00493-0Section 0218

Section 03

Robustness

Leakage, decoherence, device imperfections

Leakage

Data

Valley hotspot

✗

Valley splitting / GHz

30.00

Relative energy error

0.013

Renormalized relative error

−0.00015

C. K. Long et al., npj Quantum Inf. 11, 113 (2025), doi:10.1038/s41534-025-01027-8
C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 4Section 0320

Leakage – valley hotspot

DATA

Valley hotspot

✗

✓

Valley splitting / GHz

30.00

28.03

Relative energy error

0.013

0.668

Renormalized relative error

−0.00015

−0.01225

C. K. Long et al., npj Quantum Inf. 11, 113 (2025), doi:10.1038/s41534-025-01027-8
C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 4Section 0321

Decoherence

C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 4Section 0322

Final column

T_1=10^4T_2

Chemical accuracy

1.6\,\mathrm{mE_h}

Decoherence

DATA

T_1

100\,\text{µs}

T_2

100\,\text{µs}

C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 4Section 0323

Section 04

Scalability

Optimizability & Lie algebras

Device

H(t) = -\frac{1}{2}\mu_B B_z \sum_{i=1}^{N} g_i \sigma_z^{(i)} -\textcolor{#73407f}{\frac{1}{2}\mu_B B_x(t) \sum_{i=1}^{N} g_i \sigma_x^{(i)}} +\textcolor{#21918c}{\frac{1}{4}\sum_{i=1}^{N-1} J_i(t) \vec{\sigma}^{(i)} \cdot \vec{\sigma}^{(i+1)}}

x z y

ZeemanStatic field B_z sets each qubit’s splitting.

DriveAntenna delivers the transverse control field B_x(t).

ExchangeGate voltages tune nearest-neighbour coupling J_i(t).

C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 5Section 0425A

Device

\displaystyle H(t) = -\frac{1}{2}\mu_B B_z \sum_{i=1}^{N} g_i \sigma_z^{(i)}

\displaystyle -\frac{1}{2}\mu_B B_x(t) \sum_{i=1}^{N} g_i \sigma_x^{(i)}

\displaystyle +\frac{1}{4}\sum_{i=1}^{N-1} J_i(t) \vec{\sigma}^{(i)} \cdot \vec{\sigma}^{(i+1)}

x z y

ZeemanStatic field B_z sets each qubit’s splitting.

DriveAntenna delivers the transverse control field B_x(t).

ExchangeGate voltages tune nearest-neighbour coupling J_i(t).

C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 5Section 0425B

Device dynamic Lie algebra

C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 5Section 0426A

\mathfrak{h_l}\coloneqq\left\langle\left\{i\sum_{i=1}^nB_iZ_i,\ i\vec\sigma_j\cdot\vec\sigma_{j+1}\right\}_{j=1}^{n-1}\right\rangle_{\text{Lie}}

Device dynamic Lie algebra

C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 5Section 0426B

\mathfrak{h_l}\coloneqq\left\langle\left\{i\sum_{i=1}^nB_iZ_i,\ i\vec\sigma_j\cdot\vec\sigma_{j+1}\right\}_{j=1}^{n-1}\right\rangle_{\text{Lie}}

Lie algebra

Vector space
+
binary operation called the Lie bracket \left[\bullet,\bullet\right] (e.g., commutator)

Example

Skew Hermitian matrices form \mathfrak{su}(n)

Device dynamic Lie algebra

C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 5Section 0426C

\mathfrak{h_l}\coloneqq\left\langle\left\{i\sum_{i=1}^nB_iZ_i,\ i\vec\sigma_j\cdot\vec\sigma_{j+1}\right\}_{j=1}^{n-1}\right\rangle_{\text{Lie}}

Lie algebra

Vector space
+
binary operation called the Lie bracket \left[\bullet,\bullet\right] (e.g., commutator)

Example

Skew Hermitian matrices form \mathfrak{su}(n)

Dynamic Lie algebra

The Lie algebra generated from each indepedently controllable term in the Hamiltonian

Device dynamic Lie algebra

C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 5Section 0426D

\mathfrak{h_l}\coloneqq\left\langle\left\{i\sum_{i=1}^nB_iZ_i,\ i\vec\sigma_j\cdot\vec\sigma_{j+1}\right\}_{j=1}^{n-1}\right\rangle_{\text{Lie}}

Lie algebra

Vector space
+
binary operation called the Lie bracket \left[\bullet,\bullet\right] (e.g., commutator)

Example

Skew Hermitian matrices form \mathfrak{su}(n)

Dynamic Lie algebra

The Lie algebra generated from each indepedently controllable term in the Hamiltonian

Lie group

\exp:\mathfrak g\to G

Example

\operatorname{SU}(n)=\exp\mathfrak{su}(n)

Device dynamic Lie algebra

C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 5Section 0426E

\mathfrak{h_l}\coloneqq\left\langle\left\{i\sum_{i=1}^nB_iZ_i,\ i\vec\sigma_j\cdot\vec\sigma_{j+1}\right\}_{j=1}^{n-1}\right\rangle_{\text{Lie}}

\mathfrak{h_l}\coloneqq\left\langle\left\{i\sum_{i=1}^nB_iZ_i,\ i\vec\sigma_j\cdot\vec\sigma_k\right\}_{1\le j<k\le n-1}\right\rangle_{\text{Lie}}

Lie algebra

Vector space
+
binary operation called the Lie bracket \left[\bullet,\bullet\right] (e.g., commutator)

Example

Skew Hermitian matrices form \mathfrak{su}(n)

Dynamic Lie algebra

The Lie algebra generated from each indepedently controllable term in the Hamiltonian

Lie group

\exp:\mathfrak g\to G

Example

\operatorname{SU}(n)=\exp\mathfrak{su}(n)

Device dynamic Lie algebra

C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 5Section 0426F

\mathfrak{h_l}\coloneqq\left\langle\left\{i\sum_{i=1}^nB_iZ_i,\ i\vec\sigma_j\cdot\vec\sigma_k\right\}_{1\le j<k\le n-1}\right\rangle_{\text{Lie}}

Lie algebra

Vector space
+
binary operation called the Lie bracket \left[\bullet,\bullet\right] (e.g., commutator)

Example

Skew Hermitian matrices form \mathfrak{su}(n)

Dynamic Lie algebra

The Lie algebra generated from each indepedently controllable term in the Hamiltonian

Lie group

\exp:\mathfrak g\to G

Example

\operatorname{SU}(n)=\exp\mathfrak{su}(n)

Commutators

C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 5Section 0427A

\mathfrak{h_l}\coloneqq\left\langle\left\{i\sum_{i=1}^nB_iZ_i,\ i\vec\sigma_j\cdot\vec\sigma_k\right\}_{1\le j<k\le n-1}\right\rangle_{\text{Lie}}

\begin{aligned} \left[i\sum_{i=1}^nB_iZ_i,\ i\vec\sigma_j\cdot\vec\sigma_k\right] \end{aligned}

Lie algebra

Vector space
+
binary operation called the Lie bracket \left[\bullet,\bullet\right] (e.g., commutator)

Example

Skew Hermitian matrices form \mathfrak{su}(n)

Dynamic Lie algebra

The Lie algebra generated from each indepedently controllable term in the Hamiltonian

Lie group

\exp:\mathfrak g\to G

Example

\operatorname{SU}(n)=\exp\mathfrak{su}(n)

Commutators

C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 5Section 0427B

\mathfrak{h_l}\coloneqq\left\langle\left\{i\sum_{i=1}^nB_iZ_i,\ i\vec\sigma_j\cdot\vec\sigma_k\right\}_{1\le j<k\le n-1}\right\rangle_{\text{Lie}}

\begin{aligned} \left[i\sum_{i=1}^nB_iZ_i,\ i\vec\sigma_j\cdot\vec\sigma_k\right]&\propto i\frac{1}{2}(X_jY_k-Y_jX_k). \end{aligned}

Lie algebra

Vector space
+
binary operation called the Lie bracket \left[\bullet,\bullet\right] (e.g., commutator)

Example

Skew Hermitian matrices form \mathfrak{su}(n)

Dynamic Lie algebra

The Lie algebra generated from each indepedently controllable term in the Hamiltonian

Lie group

\exp:\mathfrak g\to G

Example

\operatorname{SU}(n)=\exp\mathfrak{su}(n)

Commutators

C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 5Section 0427C

\mathfrak{h_l}\coloneqq\left\langle\left\{i\sum_{i=1}^nB_iZ_i,\ i\vec\sigma_j\cdot\vec\sigma_k\right\}_{1\le j<k\le n-1}\right\rangle_{\text{Lie}}

\begin{aligned} \left[i\sum_{i=1}^nB_iZ_i,\ i\vec\sigma_j\cdot\vec\sigma_k\right]&\propto i\frac{1}{2}(X_jY_k-Y_jX_k),\\ \left[i\sum_{i=1}^nB_iZ_i,\ i(X_jY_k-Y_jX_k)\right] \end{aligned}

Lie algebra

Vector space
+
binary operation called the Lie bracket \left[\bullet,\bullet\right] (e.g., commutator)

Example

Skew Hermitian matrices form \mathfrak{su}(n)

Dynamic Lie algebra

The Lie algebra generated from each indepedently controllable term in the Hamiltonian

Lie group

\exp:\mathfrak g\to G

Example

\operatorname{SU}(n)=\exp\mathfrak{su}(n)

Commutators

C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 5Section 0427D

\mathfrak{h_l}\coloneqq\left\langle\left\{i\sum_{i=1}^nB_iZ_i,\ i\vec\sigma_j\cdot\vec\sigma_k\right\}_{1\le j<k\le n-1}\right\rangle_{\text{Lie}}

\begin{aligned} \left[i\sum_{i=1}^nB_iZ_i,\ i\vec\sigma_j\cdot\vec\sigma_k\right]&\propto i\frac{1}{2}(X_jY_k-Y_jX_k),\\ \left[i\sum_{i=1}^nB_iZ_i,\ i(X_jY_k-Y_jX_k)\right]&\propto i\frac{1}{2}(X_jX_k+Y_jY_k). \end{aligned}

Lie algebra

Vector space
+
binary operation called the Lie bracket \left[\bullet,\bullet\right] (e.g., commutator)

Example

Skew Hermitian matrices form \mathfrak{su}(n)

Dynamic Lie algebra

The Lie algebra generated from each indepedently controllable term in the Hamiltonian

Lie group

\exp:\mathfrak g\to G

Example

\operatorname{SU}(n)=\exp\mathfrak{su}(n)

Commutators

C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 5Section 0427E

\mathfrak{h_l}\coloneqq\left\langle\left\{i\sum_{i=1}^nB_iZ_i,\ i\vec\sigma_j\cdot\vec\sigma_k\right\}_{1\le j<k\le n-1}\right\rangle_{\text{Lie}}

Lie algebra

Vector space
+
binary operation called the Lie bracket \left[\bullet,\bullet\right] (e.g., commutator)

Example

Skew Hermitian matrices form \mathfrak{su}(n)

Dynamic Lie algebra

The Lie algebra generated from each indepedently controllable term in the Hamiltonian

Lie group

\exp:\mathfrak g\to G

Example

\operatorname{SU}(n)=\exp\mathfrak{su}(n)

Commutators

C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 5Section 0427F

\mathfrak{h_l}\coloneqq\left\langle\left\{i\sum_{i=1}^nB_iZ_i,\ i\vec\sigma_j\cdot\vec\sigma_k\right\}_{1\le j<k\le n-1}\right\rangle_{\text{Lie}}

Commutators

C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 5Section 0427G

\mathfrak{h_l}\coloneqq\left\langle\left\{i\sum_{i=1}^nB_iZ_i,\ i\vec\sigma_j\cdot\vec\sigma_k\right\}_{1\le j<k\le n-1}\right\rangle_{\text{Lie}}

Commutators

C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 5Section 0427H

\mathfrak{h_l}\coloneqq\left\langle\left\{i\sum_{i=1}^nB_iZ_i,\ i\vec\sigma_j\cdot\vec\sigma_k\right\}_{1\le j<k\le n-1}\right\rangle_{\text{Lie}}

Commutators

C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 5Section 0427I

\mathfrak{h_l}\coloneqq\left\langle\left\{i\sum_{i=1}^nB_iZ_i,\ i\vec\sigma_j\cdot\vec\sigma_k\right\}_{1\le j<k\le n-1}\right\rangle_{\text{Lie}}

\begin{aligned} \left[i\sum_{i=1}^nB_iZ_i,\ i\vec\sigma_j\cdot\vec\sigma_k\right]&\propto i\frac{1}{2}(X_jY_k-Y_jX_k)=i\begin{bmatrix}0\\&0&-i\\&i&\phantom{-}0\\&&&0\end{bmatrix}\eqqcolon i\overline Y_{jk},\\ \left[i\sum_{i=1}^nB_iZ_i,\ i(X_jY_k-Y_jX_k)\right]&\propto i\frac{1}{2}(X_jX_k+Y_jY_k)=i\begin{bmatrix}0\\&0&\phantom{-}1\\&1&\phantom{-}0\\&&&0\end{bmatrix},\\ \left[i(X_jX_k+Y_jY_k),\ i(X_jY_k-Y_jX_k)\right]&\propto i\frac{1}{2}(Z_j-Z_k). \end{aligned}

Commutators

C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 5Section 0427J

\mathfrak{h_l}\coloneqq\left\langle\left\{i\sum_{i=1}^nB_iZ_i,\ i\vec\sigma_j\cdot\vec\sigma_k\right\}_{1\le j<k\le n-1}\right\rangle_{\text{Lie}}

\begin{aligned} \left[i\sum_{i=1}^nB_iZ_i,\ i\vec\sigma_j\cdot\vec\sigma_k\right]&\propto i\frac{1}{2}(X_jY_k-Y_jX_k)=i\begin{bmatrix}0\\&0&-i\\&i&\phantom{-}0\\&&&0\end{bmatrix}\eqqcolon i\overline Y_{jk},\\ \left[i\sum_{i=1}^nB_iZ_i,\ i(X_jY_k-Y_jX_k)\right]&\propto i\frac{1}{2}(X_jX_k+Y_jY_k)=i\begin{bmatrix}0\\&0&\phantom{-}1\\&1&\phantom{-}0\\&&&0\end{bmatrix}\eqqcolon i\overline X_{jk},\\ \left[i(X_jX_k+Y_jY_k),\ i(X_jY_k-Y_jX_k)\right]&\propto i\frac{1}{2}(Z_j-Z_k). \end{aligned}

Commutators

C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 5Section 0427K

\mathfrak{h_l}\coloneqq\left\langle\left\{i\sum_{i=1}^nB_iZ_i,\ i\vec\sigma_j\cdot\vec\sigma_k\right\}_{1\le j<k\le n-1}\right\rangle_{\text{Lie}}

\begin{aligned} \left[i\sum_{i=1}^nB_iZ_i,\ i\vec\sigma_j\cdot\vec\sigma_k\right]&\propto i\frac{1}{2}(X_jY_k-Y_jX_k)=i\begin{bmatrix}0\\&0&-i\\&i&\phantom{-}0\\&&&0\end{bmatrix}\eqqcolon i\overline Y_{jk},\\ \left[i\sum_{i=1}^nB_iZ_i,\ i(X_jY_k-Y_jX_k)\right]&\propto i\frac{1}{2}(X_jX_k+Y_jY_k)=i\begin{bmatrix}0\\&0&\phantom{-}1\\&1&\phantom{-}0\\&&&0\end{bmatrix}\eqqcolon i\overline X_{jk},\\ \left[i(X_jX_k+Y_jY_k),\ i(X_jY_k-Y_jX_k)\right]&\propto i\frac{1}{2}\mathrlap{(Z_j-Z_k)}\hphantom{(X_jX_k+Y_jY_k)}=i\begin{bmatrix}0\\&1\\&&-1\\&&&0\end{bmatrix}. \end{aligned}

Commutators

C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 5Section 0427L

\mathfrak{h_l}\coloneqq\left\langle\left\{i\sum_{i=1}^nB_iZ_i,\ i\vec\sigma_j\cdot\vec\sigma_k\right\}_{1\le j<k\le n-1}\right\rangle_{\text{Lie}}

\begin{aligned} \left[i\sum_{i=1}^nB_iZ_i,\ i\vec\sigma_j\cdot\vec\sigma_k\right]&\propto i\frac{1}{2}(X_jY_k-Y_jX_k)=i\begin{bmatrix}0\\&0&-i\\&i&\phantom{-}0\\&&&0\end{bmatrix}\eqqcolon i\overline Y_{jk},\\ \left[i\sum_{i=1}^nB_iZ_i,\ i(X_jY_k-Y_jX_k)\right]&\propto i\frac{1}{2}(X_jX_k+Y_jY_k)=i\begin{bmatrix}0\\&0&\phantom{-}1\\&1&\phantom{-}0\\&&&0\end{bmatrix}\eqqcolon i\overline X_{jk},\\ \left[i(X_jX_k+Y_jY_k),\ i(X_jY_k-Y_jX_k)\right]&\propto i\frac{1}{2}\mathrlap{(Z_j-Z_k)}\hphantom{(X_jX_k+Y_jY_k)}=i\begin{bmatrix}0\\&1\\&&-1\\&&&0\end{bmatrix}\eqqcolon i\overline Z_{jk}. \end{aligned}

Device dynamic Lie algebra

C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 5Section 0428A

\mathfrak{h_l}=\left\langle\left\{i\sum_{i=1}^nZ_i,\ iZ_jZ_k,\ i\overline X_{jk},\ i\overline Y_{jk},\ i\overline Z_{jk}\right\}_{1\le j<k\le n-1}\right\rangle_{\text{Lie}}

Commutators

C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 5Section 0428B

\mathfrak{h_l}=\left\langle\left\{i\sum_{i=1}^nZ_i,\ iZ_jZ_k,\ i\overline X_{jk},\ i\overline Y_{jk},\ i\overline Z_{jk}\right\}_{1\le j<k\le n-1}\right\rangle_{\text{Lie}}

\left[i\overline Y_{ij}I_k,\ iI_i\overline Y_{jk}\otimes\left\{I, Z\right\}^{\otimes n-3}\right]

Commutators

C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 5Section 0428C

\mathfrak{h_l}=\left\langle\left\{i\sum_{i=1}^nZ_i,\ iZ_jZ_k,\ i\overline X_{jk},\ i\overline Y_{jk},\ i\overline Z_{jk}\right\}_{1\le j<k\le n-1}\right\rangle_{\text{Lie}}

\left[i\overline Y_{ij}I_k,\ iI_i\overline Y_{jk}\otimes\left\{I, Z\right\}^{\otimes n-3}\right]=\left[i\overline Y_{ij}I_k,\ iI_i\overline Y_{jk}\right]\otimes\left\{I, Z\right\}^{\otimes n-3}

Commutators

C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 5Section 0428D

\mathfrak{h_l}=\left\langle\left\{i\sum_{i=1}^nZ_i,\ iZ_jZ_k,\ i\overline X_{jk},\ i\overline Y_{jk},\ i\overline Z_{jk}\right\}_{1\le j<k\le n-1}\right\rangle_{\text{Lie}}

\left[i\overline Y_{ij}I_k,\ iI_i\overline Y_{jk}\otimes\left\{I, Z\right\}^{\otimes n-3}\right]=\left[i\overline Y_{ij}I_k,\ iI_i\overline Y_{jk}\right]\otimes\left\{I, Z\right\}^{\otimes n-3}=i\overline Y_{ik} Z_j\otimes\left\{I, Z\right\}^{\otimes n-3}

Commutators

C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 5Section 0428E

\mathfrak{h_l}=\left\langle\left\{i\sum_{i=1}^nZ_i,\ iZ_jZ_k,\ i\overline X_{jk},\ i\overline Y_{jk},\ i\overline Z_{jk}\right\}_{1\le j<k\le n-1}\right\rangle_{\text{Lie}}

\left[i\overline X_{jk},\ i\overline Y_{jk}\otimes\left\{I, Z\right\}^{\otimes n-2}\right]=\left[i\overline X_{jk},\ i\overline Y_{jk}\right]\otimes\left\{I, Z\right\}^{\otimes n-2}=2i\overline Z_{jk}\otimes\left\{I, Z\right\}^{\otimes n-2}

Commutators

C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 5Section 0428F

\mathfrak{h_l}=\left\langle\left\{i\sum_{i=1}^nZ_i,\ iZ_jZ_k,\ i\overline X_{jk},\ i\overline Y_{jk},\ i\overline Z_{jk}\right\}_{1\le j<k\le n-1}\right\rangle_{\text{Lie}}

\left[i\overline Y_{jk},\ i\overline Z_{jk}\otimes\left\{I, Z\right\}^{\otimes n-2}\right]=\left[i\overline Y_{jk},\ i\overline Z_{jk}\right]\otimes\left\{I, Z\right\}^{\otimes n-2}=2i\overline X_{jk}\otimes\left\{I, Z\right\}^{\otimes n-2}

Device dynamic Lie algebra

C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 5Section 0429

\mathfrak{h_l}\!=\!\left\langle\!\left\{\!i\!\sum_{i=1}^nZ_i,i\!\!\!\!\sum_{\substack{l,m=1\colon\\l\ne m}}^n\!\!\!\!Z_lZ_m,i\overline X_{jk}\otimes\left\{\tfrac{1}{2}(I\!\pm \!Z)\right\}^{\otimes n-2},i\overline Y_{jk}\otimes\left\{\tfrac{1}{2}(I\!\pm \!Z)\right\}^{\otimes n-2},i\overline Z_{jk}\otimes\left\{\tfrac{1}{2}(I\!\pm \!Z)\right\}^{\otimes n-2}\right\}_{\substack{1\le j<k\\\le n-1}}\right\rangle_{\!\!\!\text{Lie}}

Gray code

C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 5Section 0430A

\begin{aligned} \ket{\begin{matrix} 0&1&0&0&1 \end{matrix}}\\ \phantom{\ket{\begin{matrix} 0&1&0&0&0 \end{matrix}}}\\ \phantom{\ket{\begin{matrix} 0&1&1&0&0 \end{matrix}}}\\ \phantom{\ket{\begin{matrix} 0&0&1&0&0 \end{matrix}}}\\ \ket{\begin{matrix} 1&0&1&0&0 \end{matrix}} \end{aligned}

Gray code

C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 5Section 0430B

\begin{aligned} \ket{\begin{matrix} 0&1&0&0&1 \end{matrix}}\\ \ket{\begin{matrix} 0&1&0&0&\textcolor{#21918c}{0} \end{matrix}}\\ \phantom{\ket{\begin{matrix} 0&1&1&0&0 \end{matrix}}}\\ \phantom{\ket{\begin{matrix} 0&0&1&0&0 \end{matrix}}}\\ \ket{\begin{matrix} 1&0&1&0&0 \end{matrix}} \end{aligned}

Gray code

C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 5Section 0430C

\begin{aligned} \ket{\begin{matrix} 0&1&0&0&1 \end{matrix}}\\ \ket{\begin{matrix} 0&1&0&0&0 \end{matrix}}\\ \ket{\begin{matrix} 0&1&\textcolor{#21918c}{1}&0&0 \end{matrix}}\\ \phantom{\ket{\begin{matrix} 0&0&1&0&0 \end{matrix}}}\\ \ket{\begin{matrix} 1&0&1&0&0 \end{matrix}} \end{aligned}

Gray code

C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 5Section 0430D

\begin{aligned} \ket{\begin{matrix} 0&1&0&0&1 \end{matrix}}\\ \ket{\begin{matrix} 0&1&0&0&0 \end{matrix}}\\ \ket{\begin{matrix} 0&1&1&0&0 \end{matrix}}\\ \ket{\begin{matrix} 0&\textcolor{#21918c}{0}&1&0&0 \end{matrix}}\\ \ket{\begin{matrix} 1&0&1&0&0 \end{matrix}} \end{aligned}

Gray-inspired code

C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 5Section 0431A

\begin{aligned} \ket{\begin{matrix} 0&1&0&0&1 \end{matrix}}\equiv \ket{\overline 0}\\ \phantom{\ket{\begin{matrix} 0&1&1&0&0 \end{matrix}}\equiv \ket{\overline 1}}\\ \ket{\begin{matrix} 1&0&1&0&0 \end{matrix}}\equiv \ket{\overline 2} \end{aligned}

Gray-inspired code

C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 5Section 0431B

\begin{aligned} \ket{\begin{matrix} 0&1&0&0&1 \end{matrix}}\equiv \ket{\overline 0}\\ \ket{\begin{matrix} 0&1&\textcolor{#21918c}{1}&0&\textcolor{#21918c}{0} \end{matrix}}\equiv \ket{\overline 1}\\ \ket{\begin{matrix} 1&0&1&0&0 \end{matrix}}\equiv \ket{\overline 2} \end{aligned}

Gray-inspired code

C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 5Section 0431C

\begin{aligned} \ket{\begin{matrix} 0&1&0&0&1 \end{matrix}}\equiv \ket{\overline 0}\\ \ket{\begin{matrix} 0&1&1&0&0 \end{matrix}}\equiv \ket{\overline 1}\\ \ket{\begin{matrix} 1&0&1&0&0 \end{matrix}}\equiv \ket{\overline 2} \end{aligned}

Gray-inspired code

C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 5Section 0431D

\sum_{m=0}^ka_m\ket{\overline m}+\ket{\psi}

Gray-inspired code

C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 5Section 0431E

\sum_{m=0}^ka_m\ket{\overline m}+\ket{\psi}

\ket{\overline m}\mapsto\ket{\overline{m+1}}\text{ and }\ket{\overline{m+1}}\mapsto-\ket{\overline m}

Gray-inspired code

C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 5Section 0431F

\sum_{m=0}^ka_m\ket{\overline m}+\ket{\psi}\mapsto-\sum_{m=0}^{k-2}a_{m+1}\ket{\overline m}+a_0\ket{\overline{k-1}}+a_k\ket{\overline k}+\ket{\psi}

Gray-inspired code

C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 5Section 0431G

\sum_{m=0}^ka_m\ket{\overline m}+\ket{\psi}\mapsto-\sum_{m=0}^{k-2}a_{m+1}\ket{\overline m}+\textcolor{#21918c}{a_0'}\ket{\overline{k-1}}+\textcolor{#21918c}{a_k'}\ket{\overline k}+\ket{\psi}

\begin{bmatrix} \textcolor{#21918c}{a_0'}\\ \textcolor{#21918c}{a_k'} \end{bmatrix}=U\begin{bmatrix} a_0\\ a_k \end{bmatrix}\qquad U\in \operatorname{SU}(2)

Gray-inspired code

C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 5Section 0431H

\sum_{m=0}^ka_m\ket{\overline m}+\ket{\psi}\mapsto\sum_{m=1}^{k-1}a_m\ket{\overline m}+a_0'\ket{\overline{0}}+a_k'\ket{\overline k}+\ket{\psi}

\ket{\overline m}\mapsfrom\ket{\overline{m+1}}\text{ and }\ket{\overline{m+1}}\mapsfrom-\ket{\overline m}

Device dynamic Lie algebra

C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 5Section 0432

\mathfrak{h_l}=\left\langle\left\{i\sum_{i=1}^nZ_i,\ i\sum_{\substack{l,m=1\colon\\l\ne m}}^nZ_lZ_m\right\}\cup\bigoplus_{N_{\text{e}}=0}^n\mathfrak{su}\!\left[\binom{n}{N_{\text{e}}}\right]\right\rangle_{\text{Lie}}

Centralizer

C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 5Section 0433

i\sum_{\substack{j,k=1\colon\\j\ne k}}^nZ_jZ_k=i\left(\sum_{i=1}^nZ_i\right)^2-inI

Device dynamic Lie algebra

C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 5Section 0434

\mathfrak{h_l}=\mathfrak{u}(1)^{\oplus 2}\oplus\textcolor{#21918c}{\bigoplus_{N_{\text{e}}=0}^n\mathfrak{su}\!\left[\binom{n}{N_{\text{e}}}\right]}

Qubit-excitation dynamic Lie algebra

C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 5Section 0435

\mathfrak{h_l}=\mathfrak{u}(1)^{\oplus 2}\oplus\textcolor{#21918c}{\bigoplus_{N_{\text{e}}=0}^n\mathfrak{su}\!\left[\binom{n}{N_{\text{e}}}\right]}

\mathfrak{q}=\bigoplus_{N_{\text{e}}=0}^n\mathfrak{o}\!\left[\binom{n}{N_{\text{e}}}\right]

Barren plateaus

M. Ragone et al., Nat Commun 15, 7172 (2024), doi:10.1038/s41467-024-49909-3Section 0436A

\mathfrak{h_l}=\mathfrak{u}(1)^{\oplus 2}\oplus\textcolor{#21918c}{\bigoplus_{N_{\text{e}}=0}^n\mathfrak{su}\!\left[\binom{n}{N_{\text{e}}}\right]}

\mathfrak{q}=\bigoplus_{N_{\text{e}}=0}^n\mathfrak{o}\!\left[\binom{n}{N_{\text{e}}}\right]

Theorem (Ragone et al. informal)

The dynamic Lie algebra is \mathfrak g=\bigoplus_{i}\mathfrak g_i,
The initial state is in \mathfrak g_i,
The ansatz is sufficiently deep,

then

\operatorname{Var}_{\vec\theta}E(\vec\theta)\propto\frac{1}{\dim\mathfrak{g}_i}

Barren plateaus

C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 5Section 0436B

\mathfrak{h_l}=\mathfrak{u}(1)^{\oplus 2}\oplus\textcolor{#21918c}{\bigoplus_{N_{\text{e}}=0}^n\mathfrak{su}\!\left[\binom{n}{N_{\text{e}}}\right]}

\mathfrak{q}=\bigoplus_{N_{\text{e}}=0}^n\mathfrak{o}\!\left[\binom{n}{N_{\text{e}}}\right]

Theorem (Ragone et al.informal)

The dynamic Lie algebra is \mathfrak g=\bigoplus_{i}\mathfrak g_i,
The initial state is in \mathfrak g_i,
The ansatz is sufficiently deep,

then

\operatorname{Var}_{\vec\theta}E(\vec\theta)\propto\frac{1}{\dim\mathfrak{g}_i}

\operatorname{Var}_{\vec\theta}E(\vec\theta)\propto\frac{1}{f_{\mathfrak g_i}(\dim\mathfrak{g}_i)}

Barren plateaus

C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 5Section 0436C

\mathfrak{h_l}=\mathfrak{u}(1)^{\oplus 2}\oplus\textcolor{#21918c}{\bigoplus_{N_{\text{e}}=0}^n\mathfrak{su}\!\left[\binom{n}{N_{\text{e}}}\right]}

\mathfrak{q}=\bigoplus_{N_{\text{e}}=0}^n\mathfrak{o}\!\left[\binom{n}{N_{\text{e}}}\right]

\frac{\operatorname{Var}^{\mathfrak{q}}_{\vec\theta}E(\vec\theta)}{\operatorname{Var}^{\mathfrak{h}}_{\vec\theta}E(\vec\theta)}\sim2

Comparison table

Method	Shot acceleration	Noise robustness	Landscape variance
Gate-based VQE	1	1	1
Hardware efficient	10^2\text{--}10^{3}	\sim\!10^2\text{--}10^{3}	\exp(-n)
Hamiltonian-level VQE	10^4\text{--}10^{5}	10^4\text{--}10^{5}	0.5
+ Virtual Z pulses	slightly faster	slightly larger	0.5
Hybrid gate–Hamiltonian	10^{2}	\sim\!10^{2}	1

~ indicates the value is estimated based on the other trends

Summary37

Concrete values

Method	Shot time / \mathrm{ns}	Required T_2 / \mathrm{µs}	Landscape variance
Gate-based VQE	10^4\text{--}10^{5}	10^{5}	\propto\frac{1}{\frac{1}{2}\binom{n}{N_{\mathrm{e}}}+1}
Hardware efficient	10^{2}	\sim\!10^{2}	\propto\frac{1}{2^n+1}
Hamiltonian-level VQE	2\text{--}27	1\text{--}10	\propto\frac{1}{\binom{n}{N_{\mathrm{e}}}+1}
+ Virtual Z pulses	0.3\text{--}19	\sim\!0.1\text{--}10	\propto\frac{1}{\binom{n}{N_{\mathrm{e}}}+1}
Hybrid gate–Hamiltonian	10\text{--}10^2	\sim\!10\text{--}10^2	\propto\frac{1}{\frac{1}{2}\binom{n}{N_{\mathrm{e}}}+1}

~ indicates the value is estimated based on the other trends

Summary38

Summary

01 Hamiltonian-level algorithms can be up to 10^{5} times faster than gate-based algorithms

02 Hamiltonian-level algorithms are more robust to decoherence, leakage, and device imperfections than gate-based algorithms

03 Hamiltonian-level algorithms can be designed to scale at a penalty to runtime.

Summary39

Acknowledgements

Supervisors

Crispin H. W. Barnes
Frederico Martins
David R. M. Arvidsson-Shukur
Normann Mertig

Collaborators

Nicholas J. Mayhall
Sophia E. Economou
Edwin Barnes

DISCUSSIONS

Henrik Gothen
Adam Wajed
Kieran Dalton
Yordan S. Yordanov
Peter Yang
Djamila Hiller
Yunming Qian
Andrew Ramsey
Charles G. Smith
Hisham Amer
Kyle Sherbert
Alex Thom
Simon C. Benjamin

Slides made with Claude DesignAcknowledgements40

Availability

ARTICLES

C. K. Long. PhD thesis, Univ. of Cambridge (2026)
K. Dalton, et al. npj Quantum Inf 10, 18 (2024). doi:10.1038/s41534-024-00808-x
C. K. Long, et al. Phys. Rev. A 109, 042413 (2024). doi:10.1103/PhysRevA.109.042413
C.K. Long, et al. npj Quantum Inf 11, 113 (2025). doi:10.1038/s41534-025-01027-8
C. K. Long & C. H. W. Barnes. arXiv:2509.13453 (2025)
Gothen, H., et al. arXiv:2606.17357 (2026)

SOFTWARE

github.com/Christopher-K-Long/Minimal-state-preparation-times-for-silicon-spin-qubits-source-code
Suzuki-Trotter-Evolver.readthedocs.io
PySTE.readthedocs.io
QuGrad.readthedocs.io
QuGradLab.readthedocs.io

DATA

doi:10.5281/zenodo.15676408

Acknowledgements41

Thank you

Hamiltonian-level VQAs are up to 10^{5} times faster and more robust

Contact

ckl45@cam.ac.uk

Appendix

Backup slides

C. K. Long et al., npj Quantum Inf. 11, 113 (2025), doi:10.1038/s41534-025-01027-8
C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 3BackupB1

C. K. Long et al., npj Quantum Inf. 11, 113 (2025), doi:10.1038/s41534-025-01027-8
C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 3BackupB2

C. K. Long et al., npj Quantum Inf. 11, 113 (2025), doi:10.1038/s41534-025-01027-8
C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 4BackupB3

C. K. Long et al., npj Quantum Inf. 11, 113 (2025), doi:10.1038/s41534-025-01027-8
C. K. Long, PhD thesis, Univ. of Cambridge (2026), Ch. 4BackupB4