High-efficiency multi-photon boson sampling
Hui Wang
1,2*
, Yu He
1,2*
, Yu-Huai Li
1,2*
, Zu-En Su
1,2
, Bo Li
1,2
, He-Liang Huang
1,2
,
Xing Ding
1,2
, Ming-Cheng Chen
1,2
, Chang Liu
1,2
, Jian Qin
1,2
, Jin-Peng Li
1,2
, Yu-Ming
He
1,2,3
, Christian Schneider
3
, Martin Kamp
3
, Cheng-Zhi Peng
1,2
, Sven Höfling
1,3,4
,
Chao-Yang Lu
1,2
, and Jian-Wei Pan
1,2
1
Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern
Physics, University of Science and Technology of China, Hefei, Anhui, 230026, China
2
CAS-Alibaba Quantum Computing Laboratory, CAS Centre for Excellence in Quantum
Information and Quantum Physics, University of Science and Technology of China, China
3
Technische Physik, Physikalisches Instität and Wilhelm Conrad Röntgen-Center for Complex
Material Systems, Universitat Würzburg, Am Hubland, D-97074 Würzburg, Germany
4
SUPA, School of Physics and Astronomy, University of St. Andrews, St. Andrews KY16 9SS,
United Kingdom
* These authors contributed equally to this work
Abstract:
Boson sampling is considered as a strong candidate to demonstrate the “quantum
computational supremacy” over classical computers. However, previous proof-of-
principle experiments suffered from small photon number and low sampling rates
owing to the inefficiencies of the single-photon sources and multi-port optical
interferometers. Here, we develop two central components for high-performance
boson sampling: robust multi-photon interferometers with 99% transmission rate,
and actively demultiplexed single-photon sources from a quantum-dot-micropillar
with simultaneously high efficiency, purity and indistinguishability. We implement
and validate 3-, 4-, and 5-photon boson sampling, and achieve sampling rates of
4.96 kHz, 151 Hz, and 4 Hz, respectively, which are over 24,000 times faster than
the previous experiments. Our architecture is feasible to be scaled up to larger
number of photons and with higher rate to race against classical computers, and
might provide experimental evidence against the Extended Church-Turing Thesis.
Quantum computers
1
can in principle solve certain problems faster than classical
computers. Despite substantial progress in the past two decades
2-4
, building quantum
machines that can actually outperform classical computers for some specific tasks—an
important milestone termed as “quantum supremacy”—remained challenging. In the
quest of demonstrating the “quantum supremacy”, boson sampling, an intermediate (i.e.,
non-universal) quantum computer model proposed by Aaronson and Arkhipov
5
, has
received considerable interest as it requires much less physical resources than building
universal optical quantum computers
6
.
A quantum boson-sampling machine can be realized by sending n indistinguishable
single photons through a passive m-mode ( > ) interferometer, and sampling from
the probabilistic output distribution. Mathematically, the probability amplitude of each
output outcome is proportional to the permanent of a corresponding × submatrix,
which is strongly believed to be intractable because calculating the permanent is a so-
called #P-complete complexity problem. Note that, however, boson sampling is itself
not a #P-complete problem, i.e., cannot efficiently calculate the matrix permanent. For
a specifically defined task of sampling over the entire distribution, it is expected that a
sufficiently large quantum boson-sampling machine cannot be efficiently simulated by
the classical computers
5,7,8
. In principle, a large-scale boson-sampling machine would
constitute an effective disproof against a foundational tenet in computer science: the
Extended Church-Turing Thesis, which postulates that all realistic physical systems can
be efficiently simulated with a (classical) probabilistic Turing machine.
To this end, an experimental roadmap for demonstrating “quantum supremacy” is
to construct multi-photon boson-sampling machines with increasing number of input
photons and faster sampling rates to race against classical computers. However, the
overall performance of the previous proof-of-principle boson-sampling experiments
9-17
were critically limited due to the lack of high-quality single-photon sources and low-
loss multi-mode circuits. For example, the most commonly used pseudo-single photons
created using spontaneous parametric down-conversion
18
(SPDC) were intrinsically
probabilistic and mixed with multi-photon components. The SPDC probability was
kept small (about a few percent) in order to suppress the unwanted two-photon emission.
The frequency correlation of the SPDC photon pairs and the inefficient collection into
single-mode fibers further reduced the single-photon heralding efficiency to typically a
low level of ~1% in the previous work
9-16
(see Supplementary Information Table S1).
In addition, the boson-sampling rate was significantly reduced due to the coupling and
propagation loss in the multi-mode photonic circuits. In an attempt to solve the intrinsic
probabilistic problem of SPDC, spatial or temporal multiplexing
19,20
and scattershot
boson sampling
21
schemes were proposed and demonstrated
14
. Yet, so far, all the
previous quantum optical boson-sampling machines
9-17
have demonstrated only up to
three single photons with arbitrary input configurations and 4-6 photons in special Fock
states, and the obtained sampling rates were several orders of magnitudes too low to
even outperform some of the earliest classical computers.
Indistinguishable single photons
Scaling up boson-sampling to large number of photons and with high sampling
rates represents a non-trivial experimental challenge. Importantly, it requires high-
performance single quantum emitters
22-24
that can deterministically produce one and
only one photon under each pulsed excitation. The generated photons must
simultaneously have high single-photon purity (that is, the multi-photon probability
should be vanishingly small), high indistinguishability (that is, photons are quantum
mechanically identical to each other), and high collection efficiency into a single spatial
mode
25-27
. These three key features are compatibly combined in our experiment using
pulsed s-shell resonant excitation
28
of a single self-assembled InAs/GaAs quantum dot
embedded inside a micropillar cavity
29-31
(see Fig.1 and Supplementary Information).
At π pulse excitation with a repetition rate of 76 MHz, the quantum dot-micropillar
emits ~25.6 million polarized, resonance fluorescence single photons per second at the
output of a single-mode fiber, of which ~6.5 million are eventually detected on a silicon
single-photon detector. Considering the detector dead time of ~42 ns, the actual count
rate should be corrected to 9 MHz (Fig. 2a). This is the brightest single-photon source
reported in all physical systems to date, which are directly used—without any spectral
filtering—for the photon correlation and interference measurements, and for boson
sampling. We measure its second-order correlation, and observed
2
0.02
(0) 1)7
(g =
at
zero time delay, which confirmed the high purity of the single-photon Fock state. We
perform Hong-Ou-Mandel interference as a function of the emission time separation
between two single photons
31
. With a time separation of 13 ns and 14.7 μs, photon
indistinguishabilities of 0.939(3) and 0.900(3) are measured, respectively (see Fig. 2b
and Supplementary Information). Thanks to the pulsed resonant excitation method that
eliminates dephasings and time jitter
28
, we obtain long streams near-transform-limited
single photons that are sufficient for multi-photon experiments on a semiconductor chip
for the first time.
Efficient multi-photon source
Next, we de-multiplex the single-photon stream into different spatial modes using
fast optical switches that consist of Pockels cells (with a transmission rate >99% and
extinction ratio >100:1) and polarizing beam splitters (with an extinction ratio >1200:1).
The Pockels cells, synchronized to the pulsed laser and operated at 0.76 MHz with a
rising time of 8 ns, convert the single-photon pulse train into 3, 4, or 5 separate beams
(see Supplementary Information and Fig. S5). The largest time separation between two
de-multiplexed photons is ~1.05 μs (80 pulses), where the photon indistinguishability
remains 0.923 (Fig. 2b).
To ensure that these pulses arrive simultaneously at a multi-mode interferometer,
optical fibers of different lengths and translation stage are used to finely adjust their
arrival time. The average efficiency of the optical switches is ~84.5%, which was
mainly due to the coupling efficiency and propagation loss in the optical fibers. The
efficiency can be improved in the future using faster Pockels cells (see Supplementary
Information). Thus, we eventually obtain five separate single-photon sources with end-
user efficiencies of about 28.4%. Note the active de-multiplexing method eliminates
the common technical overhead for overcoming the inhomogeneity of independent self-
assembled quantum dots to build many identical sources.
Ultra-low-loss photonic circuit
Another important ingredient for reliable and fast boson-sampling is a multi-mode
interferometric linear optical network that is phase stable, has high transmission rate,
and can implement a Haar-random unitary matrix. While the previously demonstrated
waveguide-based photonic chips showed promise for large-scale integration
10-16
, the
coupling and propagation loss in these chips seriously limited the overall efficiencies
to ~30% so far (see Supplementary Information Table S1).
Here, we put forward a new circuit design that simultaneously combines the
stability, matrix randomness, and ultra-low transmission loss. As shown in Fig. 1 (see
also Fig. S6), a 9×9 mode interferometer is constructed with a bottom-up approach,
from individual tiny trapezoid, each optically coated with polarization-dependent beam
splitting ratios (Supplementary Information). This network consists of 36 beam splitters
and 9 mirrors, and implements a near-unitary transformation to input state (Fig. 2c, d).
Thanks to the antireflection coating, the overall transmission efficiency (from input to
output) is measured to be above 99%. By Mach-Zehnder-type coherence measurements,
the spatial-mode overlap is determined to better than 99.9%. The interferometer is
housed on a temperature-stabilized baseplate, and remains stable at least for weeks (for
a test, see Fig. S7). Such a design can be further improved
32
and scaled up to reasonably
larger dimensions, which can be sufficient for the near-term goal of demonstrating
quantum supremacy through boson sampling.
Experimental results and validation
We send three, four, and five single photons into the 9-mode interferometer, and
measure the output multi-photon events, as shown in Fig. 3. We use nine silicon single-
photon avalanche detectors (efficiency ~32%), one in each output of the interferometer,
to register the no-collision (one photon per output-mode) events, which have 84, 126,
and 126 different output distributions for the 3-, 4-, and 5-boson sampling, respectively.
A total of 446084 three-photon events (Fig. 3a), 36261 four-photon events (Fig. 3b),
and 11660 five-photon events (Fig. 3c) are obtained in accumulation time of 90s, 240s,