Jonas Neergaard-Nielsen PRO
Associate Professor @ DTU Physics, Denmark
Massive quantum advantage in learning
the properties of a noisy optical channel
Jonas Neergaard-Nielsen
DTU Physics
DFS annual meeting 2025
15 May, DTU
Quantum Computational advantage
Factorisation is pretty hard
2140324650240744961264423072839333563008614715144755017797754920881418023447140136643345519095804679610992851872470914587687396261921557363047454770520805119056493106687691590019759405693457452230589325976697471681738069364894699871578494975937497937
=
64135289477071580278790190170577389084825014742943447208116859632024532344630238623598752668347708737661925585694639798853367
×
33372027594978156556226010605355114227940760344767554666784520987023841729210037080257448673296881877565718986258036932062711
- but Shor's algorithm achieves an exponential speed-up!
Implementations of Shor's algorithm
2001:
15 = 5 × 3
2007:
15 = 5 × 3
2012:
21 = 7 × 3
2019:
35 = ? × ?...
In 2019, an attempt was made to factor the number 35 using Shor's algorithm on an IBM Q System One, but the algorithm failed because of accumulating errors.
[Wikipedia]
Quantum computational advantage
While waiting for
large-scale, fault-tolerant quantum computers,
the race is on to achieve
of (almost) useless problems...
Quantum computational advantage
Google - random circuit sampling
Arute et al., Quantum supremacy using a programmable superconducting processor, Nature 574, 505 (2019)
Quantum computational advantage
Pan et al., Solving the sampling problem of the Sycamore quantum circuits, Phys. Rev. Lett. 129, 090502 (2022)
Using our algorithm the simulation for the Sycamore circuits with n = 53 qubits and m = 20 cycles is completed in about 15 hours using 512 V100 GPUs.
Quantum computational advantage
USTC - Gaussian boson sampling
Zhong et al., Quantum computational advantage using photons,
Science 370, 1460 (2020) + Zhong et al., Phys. Rev. Lett., 127, 180502 (2021)
Quantum computational advantage
Xanadu - Gaussian boson sampling
Madsen et al., Quantum computational advantage with a programmable photonic processor, Nature 606, 75 (2022)
Quantum computational advantage
Oh et al., Classical algorithm for simulating experimental Gaussian boson sampling, Nat. Phys. 20, 1461 (2024)
Quantum computational advantage
LaRose, arXiv:2412.14703
Quantum Learning advantage
Process learning
\Lambda
probe / input data
unknown process
\{\rho_i \}_{_N}
\{x_i\}_{_N}
measurement
data
\tilde\Lambda
estimate
Very general scenario:
Learn a physical process by interacting with a known probe + measurement system.
Goal: to be able to predict future interactions, discriminate between channels, etc.
Quantum learning advantage
probe / input data
unknown process
measurement
data
estimate
Entangled probes + collective measurement can overcome limitations imposed by quantum noise, considerably improving the sample complexity (scaling of \(N\) vs. system size)
\Lambda
\{\rho_i \}_{_N}
\{x_i\}_{_N}
\tilde\Lambda
TL;DR
We learn the physical properties of a
random displacement channel
using
entanglement and collective measurement
to obtain 9 orders of magnitude
provable quantum advantage
\Lambda
\rho_i
x_i
\tilde\Lambda
\{\ \ \ \}_{_N}
\{\ \ \ \}_{_N}
Huang et al., Science 376, 1182 (2022)
Quantum learning advantage
We proved that schemes exploiting entanglement with an ancillary quantum memory can learn n-mode random displacement channels with exponentially fewer samples compared to entanglement-free schemes.
Quantum learning advantage
Random displacement channel
Random displacement channel
???
Random displacement channel
In this context: a bosonic channel
\hat\rho'=\Lambda(\hat\rho)
Input: state of
a bosonic system
Output: modified state
(possibly at a different location)
\hat\rho
(light, mechanical oscillator, microwave resonator, ...)
Random displacement channel
Credit: ESO/Y. Beletsky
Random displacement channel
Random displacement channel
Random displacement channel
phase space
Random displacement channel
phase space
Random displacement channel
phase space
quadrature variables
Random displacement channel
[\hat x,\hat p]=i\hbar
\Delta x \Delta p \ge \frac{\hbar}{2}
Random displacement channel
Random displacement channel
Random displacement channel
Random displacement channel
Random displacement channel
Random displacement channel
Phase space displacement:
- not as natural as phase/amplitude noise
- but certain processes can be mapped to displacement - tilt/shift of laser beam, tailored noise models, etc.
Here:
- theoretical and practical convenience
- fundamental interest
Random displacement channel
\hat\rho \rightarrow \underbrace{\hat D(\alpha) \hat\rho \hat D^\dagger(\alpha)}_{\Lambda(\hat\rho)}
Constant displacement:
unitary operation
Random displacement channel
Random displacement:
different for every use of the channel
Random displacement channel
Random displacement:
given by a probability distribution p(α)
\hat\rho \rightarrow \underbrace{
\int d^2\alpha \ p(\alpha)
\hat D(\alpha) \hat\rho \hat D^\dagger(\alpha)}_{\Lambda(\hat\rho)}
The task, the challenge
Learn an unknown distribution p(α)
describing the channel Λ
by probing N times
and processing the data {x\({}_i\)}
\Lambda / p(\alpha)
\{\rho_i \}_{_N}
\{x_i\}_{_N}
\tilde\Lambda / \tilde p(\alpha)
p(\alpha)
Most obvious approach:
- vacuum state as probe
- double homodyne (heterodyne) measures both x and p - but limited by vacuum noise
The task, the challenge
x
p
[\hat x,\hat p]=i\hbar
Quantum entanglement (two-mode squeezed state) improves sensitivity arbitrarily
- The probe is correlated with an un-displaced "memory"
- Joint variables commute ⇒ no quantum noise
Going quantum
x' = x₁ - x₂
p' = p₁ + p₂
[\hat x_1 - \hat x_2,\hat p_1 + \hat p_2]=0
x₁ , p₁
x₂ , p₂
Going quantum - example
Truth
Distributions with narrow features are harder to resolve
←Example p(α) with oscillations in x and p
Going quantum - example
Truth
Distributions with narrow features are harder to resolve
←Example p(α) with oscillations in x and p
Instead of p(α), we mostly consider its Fourier transform (the characteristic function) - simplifies theory a bit
\lambda(\beta) = \int d^2\alpha\ p(\alpha)e^{\alpha^\dagger\beta - \beta^\dagger\alpha}
Going quantum - example
Truth
vacuum + heterodyne
Distributions with narrow features are harder to resolve
Going quantum - example
Truth
vacuum + heterodyne
entanglement-assisted
making it spicy
So far, just a single bosonic mode:
ordinary small-factor sensitivity enhancement from squeezing
making it spicy
With n >> 1 modes, correlated noise becomes super hard to reconstruct
making it spicy
The sample complexity (how many samples needed for precise estimation) is exponential in n
N \propto e^{2n}
N \propto 3^{n}
N \propto e^{2e^{-2r}n}
- but dramatically "less exponential" with entanglement (squeezing parameter r).
Down to the lab!
Using temporal modes makes scaling to many modes easier
p(α) samples
squeezing: 5.0 dB
efficiency: 79%
Channel reconstruction
n = 16
n = 30
Channel reconstruction
~ 20 min
~ 20 Myr
Hypothesis test
# classical samples needed to get same success probability as quantum
Hypothesis test
rigorous
9-orders of mag.
quantum advantage
# classical samples needed to get same success probability as quantum
TL;DR
We learn the physical properties of a
random displacement channel
using
entanglement and collective measurement
to obtain 9 orders of magnitude
provable quantum advantage
\Lambda
\rho_i
x_i
\tilde\Lambda
\{\ \ \ \}_{_N}
\{\ \ \ \}_{_N}
Zhenghao Liu
Romain Brunel
Emil Østergaard
Oscar Cordero
Jens Nielsen
Axel Bregnsbo
Ulrik Andersen
+ Chicago / Caltech / Waterloo / KAIST collaborators:
John Preskill, Liang Jiang, Changhun Oh,
Hsin-Yuan Huang, Sisi Zhou, Yat Wong, Senrui Chen
Oh, Chen, et al., Phys. Rev. Lett. 133, 230604 (2024)
Liu, Brunel, et al., arXiv:2502.07770
Teaser: Photonic quantum computing technology
Gaussian boson sampling
Measurement-based
quantum computing
Teaser: Photonic quantum computing technology
Gaussian boson sampling
Measurement-based
quantum computing
Abhi
Renato
Donghwa
Asger
squeezing on chip
measurement-induced interferometers
GKP qubits
MBQC programming
Thanks!
Massive quantum advantage in learning the properties of a noisy optical channel
By Jonas Neergaard-Nielsen
Massive quantum advantage in learning the properties of a noisy optical channel
Presented at the annual meeting of the Danish Physical Society, 2025-05-15
