# Measuring Lorentz Violation in Weak Gravity Fields

Zonghao Li  Department of Physics, Indiana University, Bloomington, IN 47405,
USA

###### Abstract

Many new linearized coefficients for Lorentz violation are discovered in our
recent work on the construction of a generic Lorentz-violating effective field
theory in curved spacetime. The new coefficients can be constrained by
experiments in weak gravity fields. In this work, we compare experiments in
different gravitational potentials and study three types of gravity-related
experiments: free-fall, gravitational interferometer, and gravitational bound-
state experiments. First constraints on the new coefficients for Lorentz
violation are extracted from those experiments.

##  1 Lorentz violation in gravity

In recent years, Lorentz violation has been a popular topic in the search for
physics beyond the Standard Model (SM) and General Relativity (GR). The
Standard-Model Extension (SME)[1, 2] has been widely used as a comprehensive
framework to study Lorentz violation in the context of effective field theory.
The minimal terms in the Lagrange density of the SME in curved spacetime were
constructed by Kostelecký in 2004,[2] and the nonminimal terms were
systematically constructed in our recent work.[3, 4] The linearizations of
those terms in weak gravity fields were also obtained.[5] The present
contribution to the proceedings of CPT’22 studies the experimental
implications of the linearized terms with a focus on matter–gravity couplings
in weak gravity fields. This work is based on the results in Ref. [kl21b].

##  2 Potential-dependent experiments

An interesting implication of the linearized Lagrange density constructed in
our recent work[5] is that the measured SME coefficients for Lorentz violation
can depend on the gravitational potential of the laboratory. Coefficients for
Lorentz violation have been measured in many experiments under the assumption
that spacetime is flat.[6] However, these experiments are typically performed
at different elevations and hence at different gravitational potentials, so
the SME coefficients can depend on the potentials.

Taking the $b$-type coefficients as an example, we know that the term in the
Lagrange density containing the minimal $b^{\kappa}$ coefficient is ${\cal
L}\supset b^{\kappa}\overline{\psi}\gamma_{\kappa}\psi$ in flat spacetime.[1]
Adding couplings with the gravitational field, we can write the generalization
of the term in a weak gravity field as[5]

${\cal
L}\supset(b_{\textrm{asy}}^{\kappa}+(b^{\L})^{\kappa\mu\nu}h_{\mu\nu}+\cdots)\overline{\psi}\gamma_{\kappa}\psi\equiv
b_{\textrm{expt}}^{\kappa}\overline{\psi}\gamma_{\kappa}\psi,$ (1)

where $h_{\mu\nu}\equiv g_{\mu\nu}-\eta_{\mu\nu}$ is the linearized
gravitational field. In a nonrelativistic weak gravity field, $h_{\mu\nu}$ can
be approximated by $h_{00}\approx-2\phi$, $h_{0j}\approx 0$, and
$h_{jk}\approx-2\phi\delta_{jk}$, where $\phi$ is the gravitational potential.
The actual $b^{\kappa}$ coefficients measured in experiments should be the
effective value

$b_{\textrm{expt}}^{\kappa}=b_{\textrm{asy}}^{\kappa}+(b^{\L})^{\kappa\mu\nu}h_{\mu\nu}+\cdots\approx
b_{\textrm{asy}}^{\kappa}-2(b^{\L})^{\kappa\Sigma\Sigma}\phi,$ (2)

where $\Sigma\Sigma$ in the index means a summation over space and time
indices in the Sun-centered frame,[6] i.e.,
$(b^{\L})^{\kappa\Sigma\Sigma}=(b^{\L})^{\kappa TT}+(b^{\L})^{\kappa
XX}+(b^{\L})^{\kappa YY}+(b^{\L})^{\kappa ZZ}$. We see that this effective
coefficient depends on the gravitational potential. Moreover, the combination
$(b^{\L})^{\kappa\Sigma\Sigma}$ can be constrained by comparing experiments
measuring $b^{\kappa}$ at different elevations.

As an example, an experiment in Seattle constrained the
${\widetilde{b}}^{X}_{e}$ coefficient, a combination of $b^{\kappa}$ and other
SME coefficients in the electron sector,[6] as
$|{\widetilde{b}}^{X}_{e}|<3.7\times 10^{-31}$ GeV.[7] Another experiment in
Taiwan measured the same combination and got
$|{\widetilde{b}}^{X}_{e}|<3.1\times 10^{-29}$ GeV.[8] The results are
obtained at different elevations with different gravitational potentials, so
we can compare them to get a constraint on the linearized coefficient
${\widetilde{b}}_{e}^{X\Sigma\Sigma}$ as
$|{\widetilde{b}}_{e}^{X\Sigma\Sigma}|<3.2\times 10^{-15}$ GeV. Similar
analyses can be done for other coefficients using experimental data summarized
in the data tables[6] for the SME.[5] More experiments can be done to measure
SME coefficients in different gravitational potentials, and those can be used
to constrain the linearized coefficients.

##  3 Free-fall experiments

Aside from comparing results in different experiments, we can also constrain
the linearized coefficients by single gravity-related experiments. To better
analyze those experiments, we derived the nonrelativistic Hamiltonian from the
linearized Lagrange density. The Hamiltonian can be written as

$H=H_{0}+H_{\phi}+H_{\sigma\phi}+H_{g}+H_{\sigma g}+\ldots,$ (3)

where $H_{0}$ is the Hamiltonian without background fields, and other
components are the corrections from background fields. Components with
subscript $\sigma$ are spin dependent, and those without are spin independent.
The exact terms in the components can be found in Ref. [kl21b].

The Hamiltonian can modify the gravitational acceleration experienced by a
dynamical system on the Earth’s surface. The spin-dependent components permits
us to study spin–gravity couplings in the SME framework for the first time. In
this section, we use free-fall experiments to test the spin–gravity couplings.

One experiment[9] compares the effective gravitational accelerations of two
isotopes of strontium atoms, the spin-zero bosonic 88Sr and the spin-9/2
fermionic 87Sr. Unpolarized 87Sr atoms were used there, so if effective
gravitational accelerations depend on spin orientations, the measured
gravitational accelerations of 87Sr atoms should span a broader range than
those of 88Sr atoms. They found no such effect to a sensitivity of $10^{-7}$.
Analyzing the result in our framework, we get bounds on nonrelativistic SME
coefficients as

$\displaystyle\Big{|}(k^{\textrm{NR}}_{\sigma\phi})_{n}^{Z}\Big{|}$
$\displaystyle<$ $\displaystyle 1\times 10^{-4}\textrm{ GeV},$
$\displaystyle\Big{|}(k^{\textrm{NR}}_{\sigma\phi
pp})_{n}^{ZJJ}-0.4(k^{\textrm{NR}}_{\sigma\phi pp})_{n}^{ZZZ}\Big{|}$
$\displaystyle<$ $\displaystyle 5\times 10^{-2}\textrm{ GeV}^{-1},$ (4)

where the subscript $n$ means the coefficients are for neutrons, and repeated
$J$ indices mean a summation over special coordinates $J=X,Y,Z$ in the Sun-
centered frame.

Another experiment[10] compares the gravitational acceleration experienced by
87Rb atoms with different spin orientations. They found no difference to a
sensitivity of $10^{-7}$. This can be translated to constraints on
nonrelativistic coefficients as

$\displaystyle\Big{|}(k^{\textrm{NR}}_{\sigma\phi})_{p}^{Z}-0.6(k^{\textrm{NR}}_{\sigma\phi})_{e}^{Z}\Big{|}$
$\displaystyle<$ $\displaystyle 2\times 10^{-5}\textrm{ GeV},$
$\displaystyle\Big{|}(k^{\textrm{NR}}_{\sigma\phi
pp})_{p}^{ZJJ}+0.3(k^{\textrm{NR}}_{\sigma\phi pp})_{p}^{JJZ}\Big{|}$
$\displaystyle<$ $\displaystyle 7\times 10^{-3}\textrm{ GeV}^{-1},$ (5)

where the subscripts $p$ and $e$ mean proton and electron flavor,
respectively.

Another type of interesting free-fall experiment is to compare the falls of
hydrogen H and antihydrogen $\overline{\textrm{H}}$. This can provide insights
on CPT symmetry. Several groups have been designing experiments for that.[11]
A detailed theoretical analysis of the falls in our framework can be found in
Ref. [kl21b]. We expect new results from those experiments in the near future.

##  4 Gravitational interferometer experiments

Our nonrelativistic Hamiltonian can also modify the gravity-induced phase
shift in gravitational interferometer experiments. In this section, we analyze
several interferometer experiments with neutrons and use them to extract
bounds on the nonrelativistic coefficients.

The first gravitational interferometer experiment was performed by Colella,
Overhauser, and Werner (COW).[12] They used Bragg diffraction to split a
coherent neutron beam into two paths at different heights and measured the
relative gravity-induced phase shift between the two paths. Unpolarized
neutron beams were used in the experiment, so it is mainly sensitive to the
spin-independent terms in our Hamiltonian.

The effective gravitational acceleration measured in the original COW
experiment attains an accuracy of 10%. From this, we deduce a constraint on a
nonrelativistic coefficient as

$(k^{\textrm{NR}}_{\phi})_{n}<1\times 10^{-1}\textrm{ GeV},$ (6)

where $(k^{\textrm{NR}}_{\phi})_{n}$ is a spin-independent coefficient in the
neutron sector. More recent versions of the COW experiment can improve this
result.[5]

The next type of interferometer experiments we consider is the OffSpec
experiment, which uses polarized nonrelativistic neutron beams and splits the
beams by magnetic fields.[13] This is sensitive to spin–gravity couplings. The
experiment measured the effective gravitational acceleration to an accuracy of
2.5%. After some analysis,[5] we get the constraint

$\Big{|}(k^{\textrm{NR}}_{\phi})_{n}+(k^{\textrm{NR}}_{\sigma\phi})^{j}_{n}\hat{s}^{j}\Big{|}<2.5\times
10^{-2}\textrm{ GeV},$ (7)

where $(k^{\textrm{NR}}_{\phi})_{n}$ and
$(k^{\textrm{NR}}_{\sigma\phi})^{j}_{n}$ are coefficients in the neutron
sector, and $\hat{s}^{j}$ is the initial polarization direction of the neutron
beams. We expect a more precise result to be obtained from a more detailed
analysis of the experiment. Also, our understanding of spin–gravity couplings
and Lorentz violation can be further improved by future experiments using
similar setups with the OffSpec experiment. For example, the coefficients
$(k^{\textrm{NR}}_{\sigma g})^{jk}_{n}$ in our Hamiltonian can be constrained
by comparing the phase shifts between horizontally split neutron beams with
different spin orientations.

##  5 Gravitational bound-state experiments

Another application of the nonrelativistic Hamiltonian concerns gravitational
bound-state experiments,[15, 16] where the bounds states of neutrons in the
Earth’s gravitational field are measured. In those experiments, our
nonrelativistic Hamiltonian can modify the energy states by changing the
potential experienced by the neutrons. Specifically, the spin-independent
terms in the Hamiltonian shift the energy levels, and the spin-dependent terms
split the energy levels.

The first gravitation bound-state experiment[15] measured the critical heights
of the bound states, which are related to the energy levels. The precision of
the measurement is around 10%. A later experiment[16] improved the precision
to around 0.3% by measuring the transition frequencies between different
energy levels. From those results, constraints on nonrelativistic SME
coefficients are found to be[5, 17]

$\displaystyle\big{|}(k^{\textrm{NR}}_{\phi})_{n}\big{|}$ $\displaystyle<$
$\displaystyle 1\times 10^{-3}\textrm{ GeV},$
$\displaystyle\sqrt{\big{[}(k^{\textrm{NR}}_{\sigma\phi})_{n}^{J}\big{]}^{2}}$
$\displaystyle<$ $\displaystyle 8\times 10^{-3}\textrm{ GeV},$ (8)

where $(k^{\textrm{NR}}_{\phi})_{n}$ and
$(k^{\textrm{NR}}_{\sigma\phi})_{n}^{J}$ are nonrelativistic coefficients in
the neutron sector, and the square implies a summation over $J=X,Y,Z$ in the
Sun-centered frame. We expect these results to be improved by more precise
future measurements.

## Acknowledgments

This work was supported in part by the U.S. Department of Energy and by the
Indiana University Center for Spacetime Symmetries (IUCSS).

## References

  * [1] D. Colladay and V.A. Kostelecký, Phys. Rev. D 55, 6760 (1997); Phys. Rev. D 58, 116002 (1998).
  * [2] V.A. Kostelecký, Phys. Rev. D 69, 105009 (2004).
  * [3] V.A. Kostelecký and Z. Li, Phys. Rev. D 99, 056016 (2019).
  * [4] V.A. Kostelecký and Z. Li, Phys. Rev. D 103, 024059 (2021).
  * [5] V.A. Kostelecký and Z. Li, Phys. Rev. D 104, 044054 (2021).
  * [6] Data Tables for Lorentz and CPT Violation, V.A. Kostelecký and N. Russell, 2023 edition, arXiv:0801.0287v16.
  * [7] B.R. Heckel et al., Phys. Rev. D 78, 092006 (2008).
  * [8] L.-S. Hou et al., Phys. Rev. Lett. 90, 201101 (2003).
  * [9] M.G. Tarallo et al., Phys. Rev. Lett. 113, 023005 (2014).
  * [10] X.C. Duan et al.., Phys. Rev. Lett. 117, 023001 (2016).
  * [11] S. Aghion et al., Nat. Commun. 5, 4538 (2014); C. Amole et al., Phys. Rev. Lett. 112, 121102 (2014); P. Indelicato et al., Nat. Commun. 4, 1787 (2013).
  * [12] R. Colella, A.W. Overhauser, and S.A. Werner, Phys. Rev. Lett. 34, 23 (1975).
  * [13] V.-O. de Haan et al., Phys. Rev. A 89, 063611 (2014).
  * [14] V. de Haan et al., Phys. Rev. A 89, 063661 (2014).
  * [15] V.V. Nesvizhevsky et al., Nature 415, 297 (2002).
  * [16] G. Cronenberg et al., Nat. Phys. 14, 1022 (2018).
  * [17] A.N. Ivanov, M. Wellenzohn, and H. Abele, Phys. Lett. B 822 136640 (2021).


# SpinQ Gemini: a desktop quantum computer for education and research

Shi-Yao Hou  Shenzhen SpinQ Technology Co., Ltd., Shenzhen, China  College of
Physics and Electronic Engineering & Center for Computational Sciences,
Sichuan Normal University, Chengdu, China  Department of Physics, The Hong
Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
Guanru Feng  Shenzhen SpinQ Technology Co., Ltd., Shenzhen, China  Zipeng Wu
Department of Physics, The Hong Kong University of Science and Technology,
Clear Water Bay, Kowloon, Hong Kong  Department of Physics, Tsinghua
University, Beijing, China  Hongyang Zou  Shenzhen SpinQ Technology Co., Ltd.,
Shenzhen, China  Wei Shi  Shenzhen SpinQ Technology Co., Ltd., Shenzhen, China
Jinfeng Zeng  Department of Physics, The Hong Kong University of Science and
Technology, Clear Water Bay, Kowloon, Hong Kong  Chenfeng Cao  Department of
Physics, The Hong Kong University of Science and Technology, Clear Water Bay,
Kowloon, Hong Kong  Sheng Yu  Shenzhen SpinQ Technology Co., Ltd., Shenzhen,
China  Zikai Sheng  Shenzhen SpinQ Technology Co., Ltd., Shenzhen, China  Xin
Rao  Shenzhen SpinQ Technology Co., Ltd., Shenzhen, China  Bing Ren  Shenzhen
SpinQ Technology Co., Ltd., Shenzhen, China  Dawei Lu  Department of Physics,
Southern University of Science and Technology, Shenzhen, China  Junting Zou
Shenzhen SpinQ Technology Co., Ltd., Shenzhen, China  Guoxing Miao  guo-
<EMAIL_ADDRESS>Institute for Quantum Computing, University of
Waterloo, Waterloo, Ontario, Canada  Shenzhen SpinQ Technology Co., Ltd.,
Shenzhen, China  Jingen Xiang<EMAIL_ADDRESS>Shenzhen SpinQ Technology Co.,
Ltd., Shenzhen, China  Bei Zeng<EMAIL_ADDRESS>Department of Physics, The Hong
Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

(August 29, 2024)

###### Abstract

SpinQ Gemini is a commercial desktop quantum computer designed and
manufactured by SpinQ Technology. It is an integrated hardware-software
system. The first generation product with two qubits was launched in January
2020. The hardware is based on NMR spectrometer, with permanent magnets
providing $\sim 1$ T magnetic field. SpinQ Gemini operates under room
temperature ($0$-$30^{\circ}$C), highlighting its lightweight (55 kg with a
volume of $70\times 40\times 80$ cm3), cost-effective (under $50$k USD), and
maintenance-free. SpinQ Gemini aims to provide real-device experience for
quantum computing education for K-12 and at the college level. It also
features quantum control design capabilities that benefit the researchers
studying quantum control and quantum noise. Since its first launch, SpinQ
Gemini has been shipped to institutions in Canada, Taiwan and Mainland China.
This paper introduces the system of design of SpinQ Gemini, from hardware to
software. We also demonstrate examples for performing quantum computing tasks
on SpinQ Gemini, including one task for a variational quantum eigensolver of a
two-qubit Heisenberg model. The next generations of SpinQ quantum computing
devices will adopt models of more qubits, advanced control functions for
researchers with comparable cost, as well as simplified models for much lower
cost (under $5$k USD) for K-12 education. We believe that low-cost portable
quantum computer products will facilitate hands-on experience for teaching
quantum computing at all levels, well-prepare younger generations of students
and researchers for the future of quantum technologies.

###### pacs:

03.65.Ud, 03.67.Dd, 03.67.Mn

##  I Introduction

SpinQ Gemini is a commercial desktop quantum computer designed and
manufactured by SpinQ Technology [1], and the first generation product with
two qubits was launched in January 2020. It is an integrated hardware-software
system as shown in Fig. 1: the left figure shows the exterior look of the
device, with a dimension of $70\times 40\times 80$ cm3, and a weight of
$55$kg; the right figure shows the user interface software SpinQuasar.

Figure 1: The photo of Gemini (left) and the homepage of SpinQuasar (right).
The Gemini connects with a personal computer (PC) installed with SpinQuasar.
SpinQuasar provides an interface for the users to manipulate the desktop
quantum computer.

The hardware part of Gemini is based on NMR spectrometer. NMR was among the
very first systems developed for quantum computing [2, 3, 4, 5, 6, 7, 8].
Despite its limitations on scalability, a lot of pioneer research and
techniques for quantum computing were first demonstrated in NMR systems [9,
10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]. Notably, many quantum control
techniques developed in NMR can be readily applied to other quantum computing
platforms [22, 23, 8, 24, 25, 26, 27, 28, 29, 30] .

Traditional NMR quantum computing is performed on commercial spectrometers
with a superconducting magnet. Those spectrometers are expensive (almost 1
million USD), large (can be as high as $\sim$3 meters), and need to work in
specially designed labs. They also require regular liquid nitrogen and helium
refills for maintenance. These issues on cost, weight, volume and extreme
physical conditions also in general exist on other systems for quantum
computing, making them hard to be accessible for users in real life, but
instead with only possible access on cloud, such as IBM Q [31], IonQ [32] and
Rigetti 111https://www.rigetti.com/. As an example, superconducting qubits
need to work in dilution fridges which usually cost almost 1 million USD.
Similar as the NMR systems, they require special lab conditions and are not
portable. Furthermore, special training is needed for operations of dilution
fridges [34, 35, 36, 37, 38, 39, 40].

With the development of permanent magnet technology in recent years [41], it
is possible to bring down the size and cost of NMR spectrometers [42, 43, 44,
45, 46]. This then makes the NMR technology an ideal choice for building
portable quantum computers. By using a permanent magnet providing 1 T magnetic
fields, SpinQ Gemini highlights its lightweight (55 kg with a volume of
70$\times$40$\times$80 cm3) and cost-effective (under $50$k USD) features, and
maintenance-free, making it portable almost like a desktop PC.

Customised quantum algorithm circuit design and programming are supported on
SpinQ Gemini using its software SpinQuasar (Fig. 1). SpinQ Gemini also
provides demonstrations of $>10$ famous quantum algorithms, such as Deutsch
algorithm [47], Grover algorithm [48, 49], and HHL algorithm [50]. It also has
build-in teaching examples for quantum mechanic, such as Rabi oscillation
observation and decoherence time measurement. Gemini not only provides a very
friendly platform for non-specialists who aim to learn quantum computing
basics and quantum programming fast, but also serves as a powerful tool for
quantum computing related research.

In this paper, we introduce the system of the first generation SpinQ Gemini.
In Sec. II, we discuss the system design, from hardware to software. In Sec.
III, we discuss how to perform quantum computing with SpinQ Gemini. We give
two concrete examples of quantum computing tasks performed on SpinQ Gemini,
one on the measurement of geometric phase of mixed state in Sec. IV, and the
other on a variational quantum eigensolver for a two-qubit Heisenberg model in
Sec. V. A discussion on future plans of next generations products will follow
in Sec. VI.

We believe that low-cost portable quantum computer products will facilitate
hands-on experience for teaching quantum computing at all levels, well-prepare
younger generations for the future of quantum technologies. It will also be
accessible to a wider range of researchers to operate under real world
conditions for quantum computers, benefiting them for further studies on
quantum control and quantum noise.

##  II System

The overall schematic diagram is shown in Fig. 2. Gemini is composed of a PC
with SpinQuasar, a control system on the master board, a radio frequency (RF)
system, a temperature control module, a pair of two permanent magnets, a field
shimming system, and a tube of sample.

Figure 2: The overview of the schematic diagram of Gemini system and the
interface SpinQuasar. The master board equipped with an FPGA, provides the
control logic of Gemini. SpinQuasar communicates with FPGA through USB so that
the user can access Gemini. The magnets, together with the temperature control
unit and the field shimming system provide a stable static homogeneous
magnetic. The RF module provides the function required to control and measure
the qubits.

The PC with SpinQuasar and the master board together realize the algorithms
and interfaces to all the functions. The magnets provide static stable
magnetic field. The field shimming system and temperature control system
together make the field stable and homogeneous enough for nuclear magnetic
resonance as well as quantum computing. The RF system provides generation,
modulation, amplification, transmission, detection and reception of the RF
pulses so that we can control and measure the quantum system.

The PC with SpinQuasar and the master board altogether realized the software
part. The modules are shown in Fig. 3. The software SpinQuasar provide an
interface for a user to communicate with the quantum computer Gemini. The
master board, of which the core device is an FPGA, realizes all the algorithms
to control the pulses (hence control the quantum state), the temperature and
shimming of the field (hence generate a stable homogeneous field). SpinQuasar
and the master board communicate with each other through USB.

Figure 3: Software structure. The software could be divided into two parts:
one (the left block) we called SpinQuasar provides an interface for users with
access to all the functions of Gemini, the other (the right block) realizes
the algorithms and controls required for Gemini to function properly. These
two parts communicate with each other through USB.

###  II.1 SpinQuasar

The left half of Fig. 3 shows the structure of SpinQuasar. It is composed of
five modules: the quantum computing module, the NMR spectroscopy module, the
instrument calibration module, the task management module and the dynamic
library module. These modules can be easily accessed from the homepage of
SpinQuasar, as shown in the right column of Fig. 1.

The quantum computing module provides an interface of a two-qubit quantum
computer and will be discussed later in Sec. III.

The NMR spectroscopy module provides an interface of direct accessing to the
1H and 31P nuclear magnetic resonance signal of our sample. It provides the
direct control of the pulse parameters on the two nuclei. Also, it provides
both the free induction decay (FID) signals and the spectra after fast Fourier
transform (FFT). This interface provides a good demonstration of a modern FFT
based NMR spectrometer.

The instrument calibration module provides an interface for users to calibrate
the parameters of the spectrometer, such as field shimming, phase calibration,
and the temperature control for the system.

Our quantum computer also supports cloud quantum computing. For cloud quantum
computing, the tasks are managed by the task management module. Also, to
support more complicated control, such as the variational quantum eigensolver
(VQE) which requires adjusting the parameters of the pulses, we provide the
APIs for programmable control, and embedded these into the dynamic library.

###  II.2 Master board

The master board integrates the digital parts of the hardware, including an
FPGA, an analog-digital converter (ADC) and a digital-analog converter (DAC).
The digital parts, as shown in the right block of Fig. 3, altogether realize
the algorithms required to generate the RF pulse, measure the readout signal,
control the temperature and shimming. These algorithms will be described
further in the introduction of each module. The ADC converts the readout
signal from the RF part as measurement, while the DAC generate the initial RF
signal for state manipulation

###  II.3 Magnets

The permanent magnets provide a stable static homogeneous magnetic field,
which split the nuclei with spin-half into two energy levels and therefore
become a qubit. The permanent magnets are two NdFeB plates. The field
generated is $\sim$1 Tesla. The field near the center of the two magnets is
roughly homogeneous: the homogeneity generated by such magnets can reach a
level of $\sim$20 ppm. Compared with modern commercial NMR spectrometers, of
which the magnet is generated by the superconducting coil which requires a
large cryogenic storage dewar and regular refilling of liquid helium and
nitrogen, the magnets of Gemini works under room temperature, hence
maintenance free and portable. The disadvantages are that the magnitude of the
magnets could only reach about 2 T, and the magnetic field is highly sensitive
to the temperature of the magnets themselves.

###  II.4 Sample

The sample we used is Dimethylphosphite ((CH3O)2PH) molecules . The 31P and 1H
atom are connected directly and provide a two-qubit quantum processor. Both
31P and 1H nuclei have a $1/2$-spin, and therefore have two energy levels. The
Lamor frequency of 31P and 1H in 1 T magnetic field are 17.2 MHz and 42.6 MHz,
respectively. The structure and the parameters of the sample are listed in
(Fig. 4).

Figure 4: The molecule structure (left) and its parameter table (right). The J
coupling between the 1H and 31P nuclear spins is 697.4 Hz. The control pulses
are on resonance with 1H and 31P spins and thus their frequency offsets are
both 0 Hz. The spin Hamiltonian is $H_{0}=\pi
J\sigma_{z}^{H}\sigma_{z}^{P}/2$, where $J=697.4$Hz.

###  II.5 RF pulse generation

The states of the nuclei could be manipulated by irradiating electro-magnetic
waves (pulses) with frequencies close to there Larmor frequency (the physics
behind this will be described later). Since the Larmor frequencies of the two
nuclei lie in the range of RF range, an RF system is designed and manufactured
to operate the quantum state and realize the quantum gates.

Figure 5: RF pulse generation and signal readout. The upper half shows how an
arbitrary wave is generated. After the wave is generated, it is power
amplified and sent to the control coil (see Fig. 2) so that the quantum state
can be manipulated. After the RF pulse irradiation, the electro-magnetic
signal induced during spin relaxation is picked up by the coil, and then is
sent to the preamplifier. After the signal is amplified, it is sent to the ADC
and processed by the master board.

###  II.6 Temprature control

The field generated by the permanent magnets is highly sensitive to the
temperature of the permanent magnets themselves. Therefore, a temperature
control system is required to guarantee that the field does not drift
following the room temperature.

Figure 6: Temperature control. The temperature control is realized by a famous
feedback algorithm: the PID algorithm. The temperature probe picks the
temperature signal and send it to the FPGA. The FPGA decides what to do next
according to the temperature signal and then controls the power of the heating
module accordingly.

###  II.7 Field Shimming

The homogeneity of the static magnetic field generated by the permanent
magnets is $\sim$20 ppm, which is too large. To compensate this inhomogeneity,
we designed a field shimming system to reduce the homogeneity to less than
$\sim 1$ ppm. The best homogeneity could be reached to $\sim 0.3$ ppm. As a
comparison, the homogeneity of a commercial nuclear magnetic spectrometer with
a superconducting magnet is $\sim 0.01$ ppm.

Figure 7: Shimming. Currently, the field shimming is accomplished by measuring
the FWHM of the hydrogen signal. It is also a feedback algorithms by reading
the FWHM information of the hydrogen signal and adjust the current of the
shimming coils.

##  III Quantum computation

###  III.1 The spin system

Gemini contains two qubits which are the two connected 31P and 1H nuclear
spins in Dimethylphosphite ((CH4O)2PH) molecules (Fig. 4). The molecules are
placed in the center of the parallel permanent magnets. The 31P and 1H larmor
frequencies are 17.2 MHz and 42.6 MHz, respectively. The 31P spin has a T1 and
T2 of 7.2 s and 0.5 s, respectively. The 1H spin has a T1 and T2 of 4 s and
0.3 s, respectively. The J coupling between the two spins is 697.4 Hz. The
control pulses are on resonance with 1H and 31P spins and thus their frequency
offsets are both 0 Hz. The spin Hamiltonian in the rotating frame is

$\displaystyle H_{0}=2\pi
JI_{z}^{H}I_{z}^{P}=\frac{\pi}{2}J\sigma_{z}^{H}\sigma_{z}^{P},$ (1)

where $J=697.4$ Hz.

###  III.2 The gate set

Single-qubit 90 degree rotation gates can be realized using square pulses of
20us and 10us for 31P and 1H, respectively. The hardware-level pulse design
and engineering are available in later versions of Gemini which provide an
arbitrary waveform generation function to users. In the current paper, all
quantum gates are realized using square pulses which are resonant with 1H or
31P and combined with free evolution. The available quantum gates contain
single-qubit and two-qubit gates. The single-qubit gates are as follows:

$\displaystyle X=\sigma_{x}=\begin{pmatrix}0&1\\\
1&0\end{pmatrix},Y=\sigma_{y}=\begin{pmatrix}0&-i\\\
i&0\end{pmatrix},Z=\sigma_{y}=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix},$
$\displaystyle
X90=e^{-i\frac{\pi}{4}\sigma_{x}},Y90=e^{-i\frac{\pi}{4}\sigma_{y}},Z90=e^{-i\frac{\pi}{4}\sigma_{z}},$
$\displaystyle
Rx=e^{-i\frac{\alpha}{2}\sigma_{x}},Ry=e^{-i\frac{\beta}{2}\sigma_{y}},Rz=e^{-i\frac{\gamma}{2}\sigma_{z}},$
$\displaystyle H=\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\\
1&-1\end{pmatrix},I=\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}.$ (2)

Here, $\alpha$, $\beta$ and $\gamma$ are the rotation angles defined by users.
The two-qubit gates are as follows

$\displaystyle\mathrm{CX}=\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&0&1\\\
0&0&1&0\end{pmatrix},\mathrm{CY}=\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\
0&0&0&-i\\\ 0&0&i&0\end{pmatrix},$
$\displaystyle\mathrm{CZ}=\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&1&0\\\
0&0&0&-1\end{pmatrix},\mathrm{delay}=e^{-itH_{0}}=e^{-it\frac{\pi}{2}J\sigma_{z}^{H}\sigma_{z}^{P}},$
(3)

where $\mathrm{CX}$ gate is the famous control NOT ($\mathrm{CNOT}$) gate.
Here, the delay gate is a free evolution gate with the duration $t$ defined by
users. It should be noted that when $t$ is large noise plays a non-negligible
role and the action of this gate is not ideal as the form in the above
equation. The single-qubit gate fidelity is estimated to be $\sim$0.99 and the
two-qubit gate fidelity is estimated to be $\sim$0.98.

###  III.3 The pseudo-pure state

The initial state of the two-qubit system is prepared to be a pseudo-pure
state (PPS) [2]. The thermal equilibrium state of a liquid-state NMR system is
subject to Boltzmann distribution and at room temperature can be expressed as
follow:

$\displaystyle\rho_{\mathrm{eq}}=\frac{e^{-H_{s}/k_{B}T}}{\mathrm{Tr}(e^{-H_{s}/k_{B}T})}\approx\frac{1}{2^{n}}I^{\otimes
n}+\Sigma^{n}_{k=1}\frac{1}{2}\epsilon_{k}\sigma_{z}^{k}.$ (4)

Here, $H_{s}$ is the spin Hamiltonian in the lab frame, and $n$ is the number
of qubits. The part $\Sigma^{n}_{k=1}\frac{1}{2}\epsilon_{k}\sigma_{z}^{k}$
gives NMR signals. At room temperature $\epsilon\sim e^{-5}$ is small and thus
this is a highly mixed state. To implement quantum computation, researchers
[2] proposed to use PPS as the initial state which has the following form,

$\displaystyle\rho_{\mathrm{pps}}=\frac{1-\eta}{2^{n}}I^{\otimes
n}+\eta|\psi\rangle\langle\psi|.$ (5)

$|\psi\rangle$ is a pure state. The PPS above has the same unitary dynamics
and observable effects as the pure state $|\psi\rangle$ except for the factor
$\eta$. PPS is widely used in NMR quantum computation.

Gemini utilizes the relaxation method in Ref. [51] to prepare the two-qubit
PPS starting from the thermal equilibrium state. As shown in Fig. 8, the first
four pulses realize a basis permutation gate which can be expressed as

$\displaystyle U_{\mathrm{permute}}=\begin{pmatrix}-i&0&0&0\\\ 0&0&0&-i\\\
0&-1&0&0\\\ 0&0&1&0\end{pmatrix}.$ (6)

$U_{\mathrm{permute}}$ permutes the basis $|01\rangle$, $|10\rangle$ and
$|11\rangle$ and leaves $|00\rangle$ unchanged upon a phase. The relaxation
method in Ref. [51] combines $U_{\mathrm{permute}}$ and a delay after it
during which T1 relaxation takes effect. By properly choosing the number of
the repetitions of this combination and the delay time $t$, the system can
reach a state whose dominantly occupied basis is $|00\rangle$ and the other
three base have the same but smaller probability. This obtained state is a PPS
and can be used as the initial state $|00\rangle$ in NMR quantum computing.

###  III.4 Density matrix reconstruction

Gemini implements quantum state tomography [52] to reconstruct the density
matrix of the quantum state after a certain gate sequence is applied. Any two-
qubit density matrix can be expressed in the following way,

$\displaystyle\rho$ $\displaystyle=\frac{1}{4}I^{\otimes
2}+\frac{1}{4}\sum_{i,j}c_{ij}\sigma_{i}\sigma_{j},$ $\displaystyle i(j)$
$\displaystyle=x,y,z,0,\mathrm{but}\;(i,j)\neq(0,0).$ (7)

Here $\sigma_{0}=I$ is the 2$\times$2 identity matrix. To reconstruct a
density matrix, one need to measure all the $c_{ij}$ which are
$c_{ij}=\mathrm{Tr}(\rho\sigma_{i}\sigma_{j})$, in other words, the
expectation values of the Pauli matrices $\sigma_{i}\sigma_{j}$. There are
total 15 of $\sigma_{i}\sigma_{j}$. But only {$\sigma_{x}I$,
$\sigma_{x}\sigma_{z}$,$\sigma_{y}I$, $\sigma_{y}\sigma_{z}$,$I\sigma_{x}$,
$\sigma_{z}\sigma_{x}$,$I\sigma_{y}$, $\sigma_{z}\sigma_{y}$} are observables
in NMR. Additional readout pulses are needed to transform the unobservable
components to be observable. For example, by applying a readout pulse $Y90$
prior to measurement, $c_{z0}$ can be obtained, $c_{z0}=\mathrm{Tr}(Y90\rho
Y90^{\dagger}\sigma_{x}I)$. In Gemini, the reconstruction is realized by
repeating a experiment six times, each time with a different readout pulse and
observing either 31P or 1H. The readout pulses and $c_{ij}$ obtained in each
of the six repetitions are listed in Fig. 9. The reconstructed PPS has a
fidelity of higher than 0.99.

Figure 8: The pulse sequence for pseudo-pure state preparation. The first four
pulses realize a basis permutation gate. After it is a long delay within which
the natural relaxation takes effect. By properly choosing the repetition
number, $N$, and the duration of the delay, $t$, the system can be steered to
the pseudo-pure state $|00\rangle$ from the thermal equilibrium state.  Figure
9: The readout pulses, observed spins and the obtained $c_{ij}$ of the six
experiments needed to reconstruct a density matrix in the form of Eq. (7).

###  III.5 Software interface

Figure 10: The SpinQuasar interface for quantum computing. There is a quantum
circuit composer where users can drag and drop the supported quantum gates to
construct a desired circuit. The corresponding pulse sequence is shown below
the quantum circuit. There are two buttons, ’Run’ and ’Simulate’, for
activation of the experiment and the simulation, respectively. The density
matrices from the experiment and the simulation are shown in the bottom half
of the interface.

The user can use the quantum computing interface of SpinQuasar to access the
quantum computing function of Gemini (Fig. 10). The structure and flow-chart
of quantum computing is shown in Fig. 11. This system wraps up the calibrated
pulses into the quantum gates aforementioned. Users can drag the supported
gates into the circuits and press Run, the two-qubit quantum computer will
start running. The final result will be shown in the form of density matrix
which is reconstructed in the way discussed in the last section. There is also
a noiseless simulator embedded in this system so that one can easily compare
the experimental results with theoretical results.

Figure 11: Realization of the quantum computing system. It consists of the
experiment on the two-qubit processor and the numerical simulation.

##  IV Application: Measurement of geometric phase of mixed state

Gemini provides demonstrations of $>10$ famous quantum algorithms, such as
Deutsch-Jozsa algorithm [53], Grover search [48], and HHL algorithm [50]. In
this paper, we provide two more advanced examples that demonstrate Gemini’s
ability on running quantum algorithms. In this section, we will demonstrate
the measurement of the geometric phase of mixed states.

###  IV.1 Theory

Geometric phase [54] is a very important concept in quantum mechanics. It is a
type of phase gained by a system that is determined by the geometry of the
system’s evolution path. The most famous geometric phase is berry phase [55]
which is associated with cyclic adiabatic evolution. Here we use a spin half
system as an example. A spin in a magnetic field is aligned with the field and
is in the state $|0\rangle$. If the field direction changes slowly and the
adiabatic conditions are satisfied, the spin direction changes also
adiabatically and is always along the field direction. When the magnetic field
returns to its initial direction, the spin returns to its initial direction as
well. However, the spin state gains a global phase and is
$e^{i(\alpha+\beta)}|0\rangle$, where $\alpha$ is the dynamic phase and
$\beta$ is the Berry phase. $\alpha$ and $\beta$ have expressions as follows:

$\displaystyle\alpha$
$\displaystyle=-\frac{1}{\hbar}\int_{0}^{\tau}E(t)\mathrm{d}t,$ (8)
$\displaystyle\beta$ $\displaystyle=-\frac{1}{2}\Omega.$ (9)

$E(t)$ is the energy of $|0\rangle$ at the time $t$ and is determined by the
instant Hamiltonian. $\Omega$ is the solid angle enclosed by the path. If the
initial state is in $|1\rangle$, and the magnetic field changes along the same
path, then the geometric phase gained by the spin is $\Omega/2$. This is
because the spin is opposite in this case and hence its path encloses a solid
angle of $-\Omega$.

Berry phase is discussed above in the context of adiabatic evolution.
Researchers have proved that adiabatic evolution is not a necessary condition
for geometric phase [56]. Geometric phase stays the same as long as the
geometry of the evolution path stays the same, and is not affected by the
Hamiltonian that drives this evolution.

Geometric phase is believed to be robust to local noise and fluctuations of
Hamiltonian parameters because of its connection with the path geometry.
Therefore, geometric quantum computation is proposed as a candidate for fault-
tolerant quantum computation [20, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66].
Geometric phase in noisy environments is also studied. When the environment is
noisy, quantum systems are always in mixed states due to the interaction with
the environment. The work in Ref. [67] provides a definition for the geometric
phase of a mixed state: It is the phase shift of the interference oscillations
in interferometry gained by the mixed state after a unitary evolution. The
unitary evolution must satisfy the parallel transport requirement [54, 67]:
The state at any instant is in-phase with the state after an infinitesimal
time. It can be proved that the dynamical phase is 0 if the parallel transport
requirement can be satisfied. After such a unitary evolution, each eigen state
of the density matrix of the initial mixed state gains a phase denoted as
$\gamma_{n}$, and has the interference visibility $\nu_{n}$. The geometric
phase $\gamma$ of the mixed state and its interference visibility $\nu$
satisfy the following equation:

$\displaystyle\nu e^{i\gamma}=\Sigma_{n}p_{n}\nu_{n}e^{i\gamma_{n}}.$ (10)

Here, $p_{n}$ is the eigen value of the $n$th eigen state of the density
matrix.

###  IV.2 Experimental protocol

We adapt the protocol used in Ref. [68] to measure the geometric phase in
mixed states as defined in Eq. 10. A two-qubit system is used in this
protocol. The first qubit is an ancilla qubit and the second qubit is in the
mixed states whose geometric phase is to be measured. The mixed state is a mix
of $|+\rangle=\sqrt{2}(|0\rangle+|1\rangle)/2$ and
$|-\rangle=\sqrt{2}(|0\rangle-|1\rangle)/2$. The initial mixed state is:

$\displaystyle\rho(0)=\frac{1}{2}(I+\vec{\bm{r}}\cdot\vec{\bm{\sigma}})=\frac{1}{2}(I-r\sigma_{x}).$
(11)

Here $\vec{\bm{r}}$ is the Bloch vector, and $r$ is its length that
corresponds to the purity of the state. If $r$ = 1, the state is a pure state
which is $|-\rangle$. If $r$ = 0, the state is totally mixed. $|-\rangle$ and
$|+\rangle$ are the two eigen states of the above density matrix with eigen
values of $(1+r)/2$ and $(1-r)/2$. Here we steer the state along the path
(A-B-C-D-A) which encloses a solid angle $\Omega$ as shown in Fig. 12. Because
the path is made up of geodesic curves, the parallel transport requirement can
be satisfied and thus the dynamical phase is zero. The $|-\rangle$ and
$|+\rangle$ states gain geometric phases of $-\Omega/2$ and $\Omega/2$, and
change to $e^{-i\Omega/2}|-\rangle$ and $e^{i\Omega/2}|+\rangle$,
respectively. It can be proved that the interference visibility of the two
eigen states are both 1. Then the geometric phase $\gamma$ of the mixed state
satisfies

$\displaystyle\nu e^{i\gamma}$
$\displaystyle=\frac{1}{2}(1+r)e^{-i\frac{\Omega}{2}}+\frac{1}{2}(1-r)e^{i\frac{\Omega}{2}}=\cos\frac{\Omega}{2}-ir\sin\frac{\Omega}{2}$
(12) $\displaystyle\gamma$ $\displaystyle=-\tan^{-1}(r\tan\frac{\Omega}{2})$
(13)

In order to measure $\gamma$, the ancilla qubit is prepared in the state
$\sqrt{2}(|0\rangle_{a}+|1\rangle_{a})/2$, which has the density matrix
$(I+\sigma_{x}^{a})/2$. We control the two-qubit system so that when the
ancilla qubit is in $|1\rangle_{a}$, the mixed state undergoes the unitary
evolution and when the ancilla qubit is in $|0\rangle_{a}$ nothing happens.
The phases gained by $|-\rangle$ and $|+\rangle$ in the mixed state (which are
$\mp\Omega/2$) are passed to the ancilla qubit. Thus after the controlled
evolution, the state of the ancilla qubit is $\sqrt{2}(|0\rangle_{a}+e^{\mp
i\Omega/2}|1\rangle_{a})/2$. The weighted average phase gained by the ancilla
qubit has the form of Eq. (13).

Figure 12: The unitary evolution path for the mixed state. The state vector of
the initial mixed state is prepared to be along -x axis pointing to A. It
evolves along the path A-B-C-D-A to return back to point to A. $\Omega$ is the
solid angle enclosed by this path. $\theta=\Omega/4$ is the angle between the
x-y plane and either of the half paths (A-B-C or C-D-A).

Next, we discuss how to prepare a mixed state in the form of Eq. (11) from the
initial PPS state $|00\rangle$. The most used method in NMR to prepare such a
mixed state is to use a pulsed gradient field, which can dephase the spin
polarization in the x-y plane fast. However, there is no pulsed gradient field
in Gemini. Considering the time scale of dephasing caused by the static field
inhomogeneity as well as T2 is much smaller than T1, we exploit the natural
dephasing to remove the unwanted polarization. To prepare a state in Eq. 11,
first the state $(I+r\sigma_{z})/2$ is prepared from $|0\rangle$:

$\displaystyle|0\rangle=\frac{1}{2}(I+\sigma_{z})\xrightarrow{R_{x}(\cos^{-1}r)}\frac{1}{2}(I+r\sigma_{z}-\sqrt{1-r^{2}}\sigma_{y}).$
(14)

The $-\sqrt{1-r^{2}}\sigma_{y}$ part in the above equation can be removed
using natural dephasing and we can get $(I+r\sigma_{z})/2$. Then, rotate
$(I+r\sigma_{z})/2$ about $y$ axis by $-\pi/2$, we get $(I-r\sigma_{x})/2$.

The $|1\rangle_{a}$-controlled unitary can be realized using the following
sequence.

$\displaystyle R_{x}(-\theta)\to\mathrm{CZ}\to
R_{x}(2\theta-\pi)\to\mathrm{CZ}$ (15)

$\theta$ is the angle between either of the two half paths and the x-y plane,
$\theta=\Omega/4$. $R_{x}(-\theta)$ operation rotates the first half of the
path to the x$<$0 half of the x-y plane. CZ is the controlled-Z gate. When the
ancilla qubit is in $|0\rangle_{a}$, CZ does nothing; when the ancilla qubit
is in $|1\rangle_{a}$, CZ rotates the mixed state about z axis by $\pi$
counterclockwise, which means the mixed state evolves along the first half of
the path. $R_{x}(2\theta-\pi)$ rotates the second half of the path to the
x$>$0 half of the x-y plane. The CZ after it realizes the evolution of the
mixed state along the second half of the path when the ancilla qubit is
$|1\rangle_{a}$. In this way, the mixed state undergoes a closed path
evolution conditional on the $|1\rangle_{a}$ state of the ancilla qubit. CZ
can be further decomposed as

$\displaystyle R_{x}(\frac{\pi}{2})\to R_{y}(\frac{\pi}{2})\to
R_{x}(-\frac{\pi}{2})\to\frac{1}{2J},$ (16)

here $1/2J$ refers to the free evolution for a duration of $1/2J$ under the J
coupling between the two qubits. The $R_{x}(\pi/2)$ gate in CZ can be combined
with the $R_{x}(-\theta)$ and $R_{x}(2\theta-\pi)$ operations in Eq. 15 and
simplified. After this simplification, the quantum circuit is shown in Fig.
13. The first qubit is the ancilla qubit. $\phi_{1}=\pi/2-\theta$,
$\phi_{2}=2\theta-\pi/2$. The geometric phase $\gamma$ can be measured by
measuring the phase change of the ancilla qubit after implementing the circuit
in Fig. 13.

Figure 13: The experimental circuit for the measurement of the geometric phase
for mixed states.

###  IV.3 Results

Experiments with $\Omega=180^{\circ}$ and $\Omega=240^{\circ}$ are carried
out. In each situation, $r$ is chosen to be [0.26, 0.50, 0.71, 0.87, 0.97].
And for each of the $r$ values, the experiment is repeated for five times to
get a mean value of the measured phases as the result of $\gamma$ (Fig. 14).
The main error sources are the non-ideal initial mixed state and RF pulse
imperfections, such as finite pulse width. The large fluctuations in the
experimental results come from the uncertainty in fitting the NMR spectra. In
spite of those errors and imperfections in experiments, the change trend of
the geometric phase as a function of the purity and the solid angle of the
path can be observed from the results.

Figure 14: The theoretical and experimental results of the mixed state
geometric phase in two situations: $\Omega=180^{\circ}$ and
$\Omega=240^{\circ}$. For each situation, five purity values were chosen for
the initial mixed state, $r$=[0.26, 0.50, 0.71, 0.87, 0.97]. For each
experimental data point, the experiments were repeated for 5 times and the
mean value of the geometric phase was calculated. The error bar indicates the
standard deviation of the geometric phase in the 5 repetitions. The
theoretical values of the geometric phase are also shown as circles
($\Omega=180^{\circ}$) and diamonds ($\Omega=240^{\circ}$) in the figure.

##  V Application: variational quantum eigensolver

In this section, we implement a variational quantum eigensolver for a two-
qubit Heisenberg model.

###  V.1 Background

Quantum algorithms such as the Grover search [48], Shor factorization [69, 70]
and HHL [50], have proven advantages over their best known classical
counterparts. However, these algorithms cannot be efficiently implemented on
near-term quantum devices due to inevitable physical noises in quantum gates.
Variational quantum algorithms (VQA) [71, 72, 73, 74, 75, 76, 77], a class of
algorithms under the hybrid quantum-classical framework, are more promising to
have practical applications on noisy intermediate-scale quantum quantum
computers [78]. VQA use a parameterized quantum circuit to estimate the cost
function $C(\boldsymbol{\theta})$ and update $\boldsymbol{\theta}$ with a
classical optimizer. Variational quantum eigensolver (VQE) [72, 79] is a
paradigmatic example of VQA that aims to find the ground state and ground
state energy of a given Hamiltonian $H$. In this section, we will demonstrate
the experimental realization of VQE on Gemini.

###  V.2 Algorithm

In classical computaitional physics (chemistry), we usually estimate the
ground state energy of $H$ through variational approaches: parameterize a wave
function $|\psi\rangle=|\psi(\bm{\theta})\rangle$, update
$\boldsymbol{\theta}$ to minimize the expectation value
$\langle\psi(\bm{\theta})|H|\psi(\bm{\theta})\rangle$ until convergence. VQE
facilitates the above procedure with a quantum computer, the wave function is
parameterized with a quantum circuit $U(\bm{\theta})$ applied to the initial
state $|\bm{0}\rangle=|0\rangle^{\otimes n}$, we optimize $\bm{\theta}$ to
minimize the expectation value,

$E(\bm{\theta})=\langle\bm{0}|U^{\dagger}(\bm{\theta})HU(\bm{\theta})|\bm{0}\rangle.$
(17)

The classical optimizer can either be gradient-based methods like SGD, Adam,
RMSprop, BFGD, or gradient-free methods like Nelder-Mead, Powell. Hardware-
efficient ansatz [72], unitary coupled clustered ansatz [80], and Hamiltonian
variational ansatz [81, 82] are common choices for $U(\bm{\theta})$. In VQE,
the gradient can be directly estimated via the parameter-shift rule [83, 84],
i.e.,

$\frac{\partial
E(\bm{\theta})}{\partial\theta_{i}}=(\left<H\right>_{\bm{\theta}_{i}^{+}}-\left<H\right>_{\bm{\theta}_{i}^{-}})/2,$
(18)

where $\bm{\theta}_{i}^{\pm}=\bm{\theta}\pm\frac{\pi}{2}\bm{e}_{i}$,
$\bm{e}_{i}$ is the $i$-th unit vector in the parameter space. Higher order
derivatives $\frac{\partial^{2}E(\bm{\theta})}{\partial\theta^{2}_{i}}$,
$\frac{\partial^{3}E(\bm{\theta})}{\partial\theta^{3}_{i}}$, which are
required in some optimizers, can be estimated in a similar way [85].

###  V.3 Experimental protocol

In this work, we apply VQE to find the ground state of 2-qubit Heisenberg
model. The Hamiltonian is

$H_{H}=X_{1}X_{2}+Y_{1}Y_{2}+Z_{1}Z_{2},$ (19)

where $X_{j},Y_{j},Z_{j}$ are the Pauli operator on the $j$-th qubit. The
hardware efficient circuit is shown in Fig. 15.

Figure 15: The hardware efficient circuit for 2-qubit VQE. $\theta_{1}$ \-
$\theta_{4}$ are the parameters to be optimized.

We implement experiments on SpinQ Gemini and IBM Q Yorktown with initial
parameter
$\bm{\theta}=\left[10.2^{\circ},8.35^{\circ},108^{\circ},91.5^{\circ}\right]$,
learning rate $\alpha=0.25$, and carry out numerical simulations.

IBM Q Yorktown is a superconducting quantum computer with 5 qubits [86], the
structure is shown in Fig. 16. We only use the first two qubits $Q_{1}$ and
$Q_{2}$, the single-gate error rates are $1.173\times 10^{-3}$ and
$9.810\times 10^{-4}$, the readout errors are $2.280\times 10^{-2}$ and
$3.660\times 10^{-2}$, the CNOT error rate is $1.825\times 10^{-2}$.

Figure 16: The hardware structure of IBM Q Yorktown.

The experimental procedures are as follows:

  * •

Initialize the circuit parameters $\bm{\theta}$,

  * •

Estimate the derivatives of $\bm{\theta}$ via parameter-shift rule,
$\frac{\partial
E(\bm{\theta})}{\partial\theta_{i}}=(\left<H\right>_{\bm{\theta}_{i}^{+}}-\left<H\right>_{\bm{\theta}_{i}^{-}})/2$.

  * •

Update the parameters with gradient descent;
$\boldsymbol{\theta^{\prime}}=\boldsymbol{\theta}-\alpha\cdot\nabla
E(\boldsymbol{\theta})$;

  * •

Estimate the expectation value
$\langle\bm{0}|U^{\dagger}(\bm{\theta})HU(\bm{\theta})|\bm{0}\rangle$;

  * •

Repeat steps 2-4 until convergence.

###  V.4 Results and simulation

The results are shown in Fig. 17 (a). show the original result of VQE
experiment on SpinQ Gemini and IBMQ Yorktown, respectively. The ground state
energy of $H_{H}$ is -3, which is shown by the … line. SpinQ Gemini and IBM Q
Yorktown perform similar, both converge to $E(\bm{\theta})\approx-2.6$ after
enough iterations, as shown by the … line and the … line, respectively.
According to our simulations and analysis, the error for Gemini mainly comes
from the inhomogeneous of the magnetic field, while the error from IBMQ mainly
comes from the readout error.

Figure 17: Results of VQE. (a) The VQE energy with respect to each iteration
on different quantum computers and simulation. The blue dot line represents
the experimental result on SpinQ Gemini. The purple dot line reperents the
CNOT error mitigated(CEM) result on SpinQ Gemini. The red triangle line
represents the numerical simulation without gate errors. The green square line
represents the experimental result on IBM Q Yorktown. The gray square line
represents the readout error mitigated(REM) result on IBM Q Yorktown. (b)
Numerical simulation for quantum circuit in VQE experiment. The blue dot line
represents the experimental result on SpinQ Gemini. The purple dot line
reperents the CNOT error mitigated result on SpinQ Gemini. The orange triangle
line represents the energy calculate by the noisy circuit with the SpinQ
Gemini experimental parameters. The brown square line represents the energy
calculate by the ideal circuit with the SpinQ Gemini experimental parameters.

The noise in quantum computer can not be neglected. To study the noise effect
and stability of SpinQ Gemini, we construct a noise model to capture the
quantum error of the SpinQ Genimi. In the realistic noise NMR quantum device,
the basic noise channel are dephasing and amplitude damping noise. For initial
state $\rho$ of the system and the quantum circuit unitary transformation $U$,
the local noise model for single-qubit and two-qubit quantum gates can be
described by the Kraus representation

$\displaystyle\rho\rightarrow\sum_{k}E_{k}U\rho
U^{\dagger}E_{k}^{\dagger}=\sum_{k}E_{k}\rho^{\prime}E_{k}^{\dagger},$ (20)

where $E_{k}$s are the Kraus operators and $\sum_{k}E_{k}E_{k}^{\dagger}=I$.
The $E_{k}$s acting on the same single qubit and two qubits as $U$ acts on.
The amplitude damping noise can be characterized by the Kraus operators,

$K_{1}=\left(\begin{array}[]{cc}1&0\\\ 0&\sqrt{1-p}\\\
\end{array}\right),K_{2}=\left(\begin{array}[]{cc}0&\sqrt{p}\\\ 0&0\\\
\end{array}\right),$

where $p\in[0,1]$ is the probability of the noise. For amplitude damping noise
on single-qubit gate $U$, the Kraus operators $E_{k}$s in Eq. (20) run over
the set $\\{K_{1},K_{2}\\}$. The Kraus operators $E_{k}$s run over the set
$\\{K_{1},K_{2}\\}\otimes\\{K_{1},K_{2}\\}$ for two-qubit noisy gate. The
dephasing noise is characterized by the Kraus operators,

$\displaystyle K_{1}=\sqrt{1-p}I_{2},\quad K_{2}=\sqrt{p}\sigma_{Z},$ (21)

where $I_{2}$ is the two dimensional identity matrix and $\sigma_{Z}$ is Pauli
operator. For dephasing noise on single-qubit gate $U$, the Kraus operators
$E_{k}$s in Eq. (20) run over the set $\\{K_{1},K_{2}\\}$. The Kraus operators
$E_{k}$s run over the set $\\{K_{1},K_{2}\\}\otimes\\{K_{1},K_{2}\\}$ for two-
qubit noisy gate.

We model the noise consisting of single qubit thermal relaxation error and two
qubit thermal relaxation error. The thermal relaxation error model applies the
amplitude damping noise after dephasing noise in each one- or two-qubit gate.
This thermal relaxation error model is characterized through the parameters
($T_{1},T_{2}^{*},t_{q}$) and the noise probability is formulated by

$\displaystyle p_{damping}$ $\displaystyle=$ $\displaystyle
1-e^{-\frac{t_{q}}{T_{1}}},$ (22) $\displaystyle p_{dephasing}$
$\displaystyle=$ $\displaystyle\frac{1}{2}\left(1-e^{-2\gamma}\right),$ (23)

where $\gamma=\frac{t_{q}}{T_{2}^{*}}-\frac{t_{q}}{2T_{1}}$. When the thermal
relaxation error model apply to single qubit gate, $t_{q}={t_{1q}}$ and
$t_{q}={t_{2q}}$ for two qubit gate. The final noise model to approximate the
noise of NMR quantum device is characterized by the parameters
$\\{T_{1},T_{2}^{*},t_{1q},t_{2q}\\}$. We set
$\\{T_{1}=5.6s,T_{2}^{*}=0.025s,t_{1q}=25\mu s,t_{2q}=800\mu s\\}$ in the
noise simulation for the NMR quantum computer. In NMR system, the dephasing
effect is caused by both the spin relaxation and the field inhomogeneity.
$T_{2}$ is used to measure the spin relaxation rate, while $T_{2}^{*}$ is used
to measure the field inhomogeneity. The $T_{2}$ data is measured using the
technique called spin echo, which can refocus the magnetisation and remove the
effect of inhomogeneous field. In our VQE experiment, we did not use such
technique, so we use $T_{2}^{*}$ instead of $T_{2}$.

With the noise model described above, we first record every parameters
$\bm{\theta}$ in each iteration of the SpinQ Gemini VQE experiment. Then we
take these parameters $\bm{\theta}$ as the parameters of quantum circuit
ansatz (Fig. 15) and calculate the energy of the Hamiltonian with respect to
the ideal circuit and noisy circuit output in each iteration. As shown in Fig.
17 (b) , the noisy circuit result shows great consistancy to the experiment
data. The paramaters $\bm{\theta}$ found by SpinQ Gemini is close to the
parameters for ground state. These results indicate that our desktop quantum
computer can run VQE algorithm well.

###  V.5 Error mitigation

Quantum error mitigation [87, 88, 89] is a technique to diminish the influence
of errors from the statistical perspective.

From the comparison and the simulation described above, we can see that the
dephasing error caused by the inhomogeneous magnetic field is dominant. Our
circuit consists of four single qubits rotations and one CNOT gate. The time
for a CNOT gate is about 800 $\mu$s and for single qubit gates is $\sim 20$
$\mu$s. Therefore, the imperfections of the CNOT gate causes primary error.
Consider the error model:

$\displaystyle\rho\rightarrow\rho_{f}=\sum_{k}E_{k}U\rho
U^{\dagger}E_{k}^{\dagger}=\sum_{k}E_{k}\rho^{\prime}E_{k}^{\dagger},$ (24)

where $E_{k}$’s are the Kraus operators, $\rho^{\prime}$ is the ideal density
matrix, and $\rho_{f}$ is the measured density matrix. Error mitigation is a
procedure that for a given $\rho_{f}$ obtained from the experiment, find a
density matrix $\rho_{0}$, which is as close to $\rho^{\prime}$ as possible,
so that the final experiment result could be improved. Here, we employ the
superoperator formalism to obtain $\rho_{0}$. This formalism works as follows.
First, let us rewrite the density matrix $\rho^{\prime}$ from an $n\times n$
matrix into an $n^{2}\times 1$ vector $\bm{\rho}^{\prime}$:

$\rho^{\prime}=\sum_{ij}\rho^{\prime}_{ij}|i\rangle\langle
j|\rightarrow\bm{\rho}^{\prime}=\sum_{ij}\rho^{\prime}_{ij}|i\rangle|j\rangle.$
(25)

Then the final state $\bm{\rho}_{f}$, which is also an $n^{2}\times 1$ vector
is

$\bm{\rho}_{f}=\hat{\hat{S}}\bm{\rho}^{\prime},$ (26)

where $\hat{\hat{S}}$ is the superoperator. With known Kraus operators
$E_{k}$, it can be obtained as

$\hat{\hat{S}}=\sum_{k}E_{k}\otimes E_{k}^{\dagger}.$ (27)

Therefore, with known $\bm{\rho}_{f}$ and $\hat{\hat{S}}$, we can get

$\bm{\rho}^{\prime}=\hat{\hat{S}}^{-1}\bm{\rho}_{f}.$ (28)

The original result and the mitigated result of Gemini is shown in Fig. 17(b).
The blue line and the purple line show the original result and the CNOT error
mitigated result. We can see that the error mitigated result of the ground
state of $H_{H}$ could reach about $-2.98$, much closer to the ideal result.
With both the simulation result and the error mitigated result, we can see
that the error model we used is a good approximation.

For the IBMQ devices, the readout errors are dominant. Here we consider the
simplest linear algebra measurement error mitigation scheme. On IBMQ Santiago
we do projective measurement and obtain one of the strings $\\{0,1\\}^{\otimes
2}$. Through tomography of measurement process, we get the probability of
string $S_{j}$ becoming $S_{k}$, denoted by $P_{kj}$. Suppose we repeat the
same measurement many times and have the string probability distribution
$C_{\text{noisy }}$, then

$C_{\text{mitigated}}=P^{-1}C_{\text{noisy }}$ (29)

provides the probability distribution with measurement error mitigated,
although $P^{-1}$ is not a physical operation. Measurement error mitigation
can efficiently improve the performance of VQE on IBMQ Santiago, as shown in
Fig. 17(a).

##  VI Discussion

For the next generations of SpinQ desktop quantum computer products, we will
develop products running with more qubits (3$\sim$4). Currently, the design of
a $3$-qubit machine is underway and the product is expected to be released in
the second quarter of 2021, with a comparable price as SpinQ Gemini (i.e.
under $50$k USD). Along the way, compatible software modules with advanced
pulse control functions will also be developed, providing more powerful
abilities for quantum algorithm/control/error mitigation designs to meet the
research needs of advanced users. Meanwhile, another direction is to make a
simplified version of the current model, making it more portable and much
lower cost (under $5$k USD). This simplified version is expected to be
released in the fourth quarter of 2021, such that it can be more affordable
for most K-12 schools around the world.

## Acknowledgement

We thank Jun Li and Tao Xin for their contribution to the early stage of this
project.

## References

  * pat [2020] Desktop nuclear magnetic resonance quantum computer (China Patent ZL 2020 2 1195611.3, 2020).
  * Cory  _et al._ [1997] D. G. Cory, A. F. Fahmy, and T. F. Havel, Ensemble quantum computing by nmr spectroscopy, Proceedings of the National Academy of Sciences 94, 1634 (1997), https://www.pnas.org/content/94/5/1634.full.pdf .
  * Gershenfeld and Chuang [1997] N. A. Gershenfeld and I. L. Chuang, Bulk spin-resonance quantum computation, Computation,” Science 275, 350 (1997).
  * Chuang  _et al._ [1998a] I. L. Chuang, N. Gershenfeld, M. G. Kubinec, and D. W. Leung, Bulk quantum computation with nuclear magnetic resonance: theory and experiment, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 447 (1998a), https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.1998.0170 .
  * Cory  _et al._ [1998a] D. G. Cory, M. D. Price, and T. F. Havel, Nuclear magnetic resonance spectroscopy: An experimentally accessible paradigm for quantum computing, Physica D: Nonlinear Phenomena 120, 82 (1998a), proceedings of the Fourth Workshop on Physics and Consumption.
  * Knill and Laflamme [1998] E. Knill and R. Laflamme, Power of one bit of quantum information, Phys. Rev. Lett. 81, 5672 (1998).
  * Cory  _et al._ [2000] D. Cory, R. Laflamme, E. Knill, L. Viola, T. Havel, N. Boulant, G. Boutis, E. Fortunato, S. Lloyd, R. Martinez, C. Negrevergne, M. Pravia, Y. Sharf, G. Teklemariam, Y. Weinstein, and W. Zurek, Nmr based quantum information processing: Achievements and prospects, Fortschritte der Physik 48, 875 (2000).
  * Vandersypen and Chuang [2005] L. M. K. Vandersypen and I. L. Chuang, Nmr techniques for quantum control and computation, Rev. Mod. Phys. 76, 1037 (2005).
  * Chuang  _et al._ [1998b] I. L. Chuang, L. M. K. Vandersypen, X. Zhou, D. W. Leung, and S. Lloyd, Experimental realization of a quantum algorithm, Nature 393, 143 (1998b).
  * Jones  _et al._ [1998] J. A. Jones, M. Mosca, and R. H. Hansen, Implementation of a quantum search algorithm on a quantum computer, Nature 393, 344 (1998).
  * Nielsen  _et al._ [1998] M. A. Nielsen, E. Knill, and R. Laflamme, Complete quantum teleportation using nuclear magnetic resonance, Nature 396, 52 (1998).
  * Somaroo  _et al._ [1999] S. Somaroo, C. H. Tseng, T. F. Havel, R. Laflamme, and D. G. Cory, Quantum simulations on a quantum computer, Phys. Rev. Lett. 82, 5381 (1999).
  * Knill  _et al._ [2000] E. Knill, R. Laflamme, R. Martinez, and C. H. Tseng, An algorithmic benchmark for quantum information processing, Nature 404, 368 (2000).
  * Vandersypen  _et al._ [2000] L. M. K. Vandersypen, M. Steffen, G. Breyta, C. S. Yannoni, R. Cleve, and I. L. Chuang, Experimental realization of an order-finding algorithm with an nmr quantum computer, Phys. Rev. Lett. 85, 5452 (2000).
  * Weinstein  _et al._ [2001] Y. S. Weinstein, M. A. Pravia, E. M. Fortunato, S. Lloyd, and D. G. Cory, Implementation of the quantum fourier transform, Phys. Rev. Lett. 86, 1889 (2001).
  * Cory  _et al._ [1998b] D. G. Cory, M. D. Price, W. Maas, E. Knill, R. Laflamme, W. H. Zurek, T. F. Havel, and S. S. Somaroo, Experimental quantum error correction, Phys. Rev. Lett. 81, 2152 (1998b).
  * Knill  _et al._ [2001] E. Knill, R. Laflamme, R. Martinez, and C. Negrevergne, Benchmarking quantum computers: The five-qubit error correcting code, Phys. Rev. Lett. 86, 5811 (2001).
  * Vandersypen  _et al._ [2001] L. M. K. Vandersypen, M. Steffen, G. Breyta, C. S. Yannoni, M. H. Sherwood, and I. L. Chuang, Experimental realization of shor’s quantum factoring algorithm using nuclear magnetic resonance, Nature 414, 883 (2001).
  * Linden  _et al._ [1998] N. Linden, H. Barjat, and R. Freeman, An implementation of the deutsch–jozsa algorithm on a three-qubit nmr quantum computer, Chemical Physics Letters 296, 61 (1998).
  * Jones  _et al._ [2000] J. A. Jones, V. Vedral, A. Ekert, and G. Castagnoli, Geometric quantum computation using nuclear magnetic resonance, Nature 403, 869–871 (2000).
  * Steffen  _et al._ [2003] M. Steffen, W. van Dam, T. Hogg, G. Breyta, and I. Chuang, Experimental implementation of an adiabatic quantum optimization algorithm, Phys. Rev. Lett. 90, 067903 (2003).
  * Carr and Purcell [1954] H. Y. Carr and E. M. Purcell, Effects of diffusion on free precession in nuclear magnetic resonance experiments, Phys. Rev. 94, 630 (1954).
  * Meiboom and Gill [1958] S. Meiboom and D. Gill, Modified spin-echo method for measuring nuclear relaxation times, Review of Scientific Instruments 29, 688 (1958), https://doi.org/10.1063/1.1716296 .
  * Khaneja  _et al._ [2005] N. Khaneja, T. Reiss, C. Kehlet, T. Schulte-Herbrüggen, and S. J. Glaser, Optimal control of coupled spin dynamics: design of nmr pulse sequences by gradient ascent algorithms, Journal of Magnetic Resonance 172, 296 (2005).
  * Schulte-Herbrüggen  _et al._ [2005] T. Schulte-Herbrüggen, A. Spörl, N. Khaneja, and S. J. Glaser, Optimal control-based efficient synthesis of building blocks of quantum algorithms: A perspective from network complexity towards time complexity, Phys. Rev. A 72, 042331 (2005).
  * Fortunato  _et al._ [2002] E. M. Fortunato, M. A. Pravia, N. Boulant, G. Teklemariam, T. F. Havel, and D. G. Cory, Design of strongly modulating pulses to implement precise effective hamiltonians for quantum information processing, The Journal of Chemical Physics 116, 7599 (2002), https://doi.org/10.1063/1.1465412 .
  * Wimperis [1994] S. Wimperis, Broadband, narrowband, and passband composite pulses for use in advanced nmr experiments, Journal of Magnetic Resonance, Series A 109, 221 (1994).
  * Brown  _et al._ [2004] K. R. Brown, A. W. Harrow, and I. L. Chuang, Arbitrarily accurate composite pulse sequences, Phys. Rev. A 70, 052318 (2004).
  * Alway and Jones [2007] W. G. Alway and J. A. Jones, Arbitrary precision composite pulses for nmr quantum computing, Journal of Magnetic Resonance 189, 114 (2007).
  * Souza  _et al._ [2011] A. M. Souza, G. A. Álvarez, and D. Suter, Robust dynamical decoupling for quantum computing and quantum memory, Phys. Rev. Lett. 106, 240501 (2011).
  * Cross [2018] A. Cross, The ibm q experience and qiskit open-source quantum computing software, in  _APS March Meeting Abstracts_ , Vol. 2018 (2018) pp. L58–003.
  * [32] C. Monroe, Ionq quantum computers: Clear to scale, Bulletin of the American Physical Society .
  * Note [1] Https://www.rigetti.com/.
  * Kjaergaard  _et al._ [2020] M. Kjaergaard, M. E. Schwartz, J. Braumüller, P. Krantz, J. I.-J. Wang, S. Gustavsson, and W. D. Oliver, Superconducting qubits: Current state of play, Annual Review of Condensed Matter Physics 11, 369 (2020).
  * Krantz  _et al._ [2019] P. Krantz, M. Kjaergaard, F. Yan, T. P. Orlando, S. Gustavsson, and W. D. Oliver, A quantum engineer’s guide to superconducting qubits, Applied Physics Reviews 6, 021318 (2019).
  * Wendin [2017] G. Wendin, Quantum information processing with superconducting circuits: a review, Reports on Progress in Physics 80, 106001 (2017).
  * Gu  _et al._ [2017] X. Gu, A. F. Kockum, A. Miranowicz, Y.-x. Liu, and F. Nori, Microwave photonics with superconducting quantum circuits, Physics Reports 718, 1 (2017).
  * You and Nori [2006] J. You and F. Nori, Superconducting circuits and quantum information, Phys. Today 58, 42 (2006).
  * You and Nori [2011] J. You and F. Nori, Atomic physics and quantum optics using superconducting circuits, Nature 474, 589 (2011).
  * Wendin and Shumeiko [2007] G. Wendin and V. Shumeiko, Quantum bits with josephson junctions, Low Temperature Physics 33, 724 (2007).
  * Brown  _et al._ [2002] D. Brown, B.-M. Ma, and Z. Chen, Developments in the processing and properties of ndfeb-type permanent magnets, Journal of Magnetism and Magnetic Materials 248, 432 (2002).
  * Calin and Helerea [2011] M. Calin and E. Helerea, Temperature influence on magnetic characteristics of ndfeb permanent magnets, in  _2011 7TH INTERNATIONAL SYMPOSIUM ON ADVANCED TOPICS IN ELECTRICAL ENGINEERING (ATEE)_ (2011) pp. 1–6.
  * [43] https://magritek.com/products/spinsolve/.
  * [44] https://nmr.oxinst.com/x-pulse.
  * [45] https://www.nanalysis.com/nmready-60pro.
  * [46] https://www.aiinmr.com/.
  * David and Richard [1992] D. David and J. Richard, Rapid solution of problems by quantum computation, Proc. R. Soc. Lond. A 439, 553–558 (1992).
  * Grover [1996] L. K. Grover, A fast quantum mechanical algorithm for database search, in  _Proceedings of the twenty-eighth annual ACM symposium on Theory of computing_ (1996) pp. 212–219.
  * Long [2001] G. L. Long, Grover algorithm with zero theoretical failure rate, Phys. Rev. A 64, 022307 (2001).
  * Harrow  _et al._ [2009] A. W. Harrow, A. Hassidim, and S. Lloyd, Quantum algorithm for linear systems of equations, Physical review letters 103, 150502 (2009).
  * Li  _et al._ [2016] J. Li, D. Lu, Z. Luo, R. Laflamme, X. Peng, and J. Du, Approximation of reachable sets for coherently controlled open quantum systems: Application to quantum state engineering, Phys. Rev. A 94, 012312 (2016).
  * Lee [2002] J.-S. Lee, The quantum state tomography on an nmr system, Physics Letters A 305, 349 (2002).
  * Deutsch and Jozsa [1992] D. Deutsch and R. Jozsa, Rapid solution of problems by quantum computation, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences 439, 553 (1992).
  * Anandan [1992] J. Anandan, The geometric phase, Nature 360, 307–313 (1992).
  * Berry [1984] M. V. Berry, Quantal phase factors accompanying adiabatic changes, Proc. R. Soc. Lond. A 392, 45–57 (1984).
  * Aharonov and Anandan [1987] Y. Aharonov and J. Anandan, Phase change during a cyclic quantum evolution, Phys. Rev. Lett. 58, 1593 (1987).
  * Zanardi and Rasetti [1999] P. Zanardi and M. Rasetti, Holonomic quantum computation, Physics Letters A 264, 94 (1999).
  * Duan  _et al._ [2001] L.-M. Duan, J. I. Cirac, and P. Zoller, Geometric manipulation of trapped ions for quantum computation, Science 292, 1695 (2001), https://science.sciencemag.org/content/292/5522/1695.full.pdf .
  * Zhu and Wang [2003] S.-L. Zhu and Z. D. Wang, Unconventional geometric quantum computation, Phys. Rev. Lett. 91, 187902 (2003).
  * Wu  _et al._ [2005] L.-A. Wu, P. Zanardi, and D. A. Lidar, Holonomic quantum computation in decoherence-free subspaces, Phys. Rev. Lett. 95, 130501 (2005).
  * Oreshkov  _et al._ [2009] O. Oreshkov, T. A. Brun, and D. A. Lidar, Fault-tolerant holonomic quantum computation, Phys. Rev. Lett. 102, 070502 (2009).
  * Xiang-Bin and Keiji [2001] W. Xiang-Bin and M. Keiji, Nonadiabatic conditional geometric phase shift with nmr, Phys. Rev. Lett. 87, 097901 (2001).
  * Sjöqvist  _et al._ [2012] E. Sjöqvist, D. M. Tong, L. M. Andersson, B. Hessmo, M. Johansson, and K. Singh, Non-adiabatic holonomic quantum computation, New Journal of Physics 14, 103035 (2012).
  * Xu  _et al._ [2012] G. F. Xu, J. Zhang, D. M. Tong, E. Sjöqvist, and L. C. Kwek, Nonadiabatic holonomic quantum computation in decoherence-free subspaces, Phys. Rev. Lett. 109, 170501 (2012).
  * Feng  _et al._ [2013] G. Feng, G. Xu, and G. Long, Experimental realization of nonadiabatic holonomic quantum computation, Phys. Rev. Lett. 110, 190501 (2013).
  * Zu  _et al._ [2014] C. Zu, W.-B. Wang, L. He, W.-G. Zhang, C.-Y. Dai, F. Wang, and L.-M. Duan, Experimental realization of universal geometric quantum gates with solid-state spins, Nature 514, 72–75 (2014).
  * Sjöqvist  _et al._ [2000] E. Sjöqvist, A. K. Pati, A. Ekert, J. S. Anandan, M. Ericsson, D. K. L. Oi, and V. Vedral, Geometric phases for mixed states in interferometry, Phys. Rev. Lett. 85, 2845 (2000).
  * Du  _et al._ [2003] J. Du, P. Zou, M. Shi, L. C. Kwek, J.-W. Pan, C. H. Oh, A. Ekert, D. K. L. Oi, and M. Ericsson, Observation of geometric phases for mixed states using nmr interferometry, Phys. Rev. Lett. 91, 100403 (2003).
  * Shor [1994] P. W. Shor, Algorithms for quantum computation: discrete logarithms and factoring, in  _Proceedings 35th Annual Symposium on Foundations of Computer Science_ (1994) pp. 124–134.
  * Shor [1999] P. W. Shor, Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer, SIAM review 41, 303 (1999).
  * Cerezo  _et al._ [2020] M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R. McClean, K. Mitarai, X. Yuan, L. Cincio, _et al._ , Variational quantum algorithms, arXiv preprint arXiv:2012.09265 (2020).
  * Kandala  _et al._ [2017] A. Kandala, A. Mezzacapo, K. Temme, M. Takita, M. Brink, J. M. Chow, and J. M. Gambetta, Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets, Nature 549, 242 (2017).
  * LaRose  _et al._ [2019] R. LaRose, A. Tikku, É. O’Neel-Judy, L. Cincio, and P. J. Coles, Variational quantum state diagonalization, npj Quantum Information 5, 1 (2019).
  * Zeng  _et al._ [2020] J. Zeng, C. Cao, C. Zhang, P. Xu, and B. Zeng, A variational quantum algorithm for hamiltonian diagonalization, arXiv preprint arXiv:2008.09854 (2020).
  * Romero  _et al._ [2017] J. Romero, J. P. Olson, and A. Aspuru-Guzik, Quantum autoencoders for efficient compression of quantum data, Quantum Science and Technology 2, 045001 (2017).
  * Cao and Wang [2020] C. Cao and X. Wang, Noise-assisted quantum autoencoder, arXiv preprint arXiv:2012.08331 (2020).
  * Yuan  _et al._ [2019] X. Yuan, S. Endo, Q. Zhao, Y. Li, and S. C. Benjamin, Theory of variational quantum simulation, Quantum 3, 191 (2019).
  * Preskill [2018] J. Preskill, Quantum computing in the nisq era and beyond, Quantum 2, 79 (2018).
  * Hempel  _et al._ [2018] C. Hempel, C. Maier, J. Romero, J. McClean, T. Monz, H. Shen, P. Jurcevic, B. P. Lanyon, P. Love, R. Babbush, _et al._ , Quantum chemistry calculations on a trapped-ion quantum simulator, Physical Review X 8, 031022 (2018).
  * Lee  _et al._ [2018] J. Lee, W. J. Huggins, M. Head-Gordon, and K. B. Whaley, Generalized unitary coupled cluster wave functions for quantum computation, Journal of chemical theory and computation 15, 311 (2018).
  * Wecker  _et al._ [2015] D. Wecker, M. B. Hastings, and M. Troyer, Progress towards practical quantum variational algorithms, Physical Review A 92, 042303 (2015).
  * Wiersema  _et al._ [2020] R. Wiersema, C. Zhou, Y. de Sereville, J. F. Carrasquilla, Y. B. Kim, and H. Yuen, Exploring entanglement and optimization within the hamiltonian variational ansatz, PRX Quantum 1, 020319 (2020).
  * Mitarai  _et al._ [2018] K. Mitarai, M. Negoro, M. Kitagawa, and K. Fujii, Quantum circuit learning, Phys. Rev. A 98, 032309 (2018).
  * Schuld  _et al._ [2019] M. Schuld, V. Bergholm, C. Gogolin, J. Izaac, and N. Killoran, Evaluating analytic gradients on quantum hardware, Phys. Rev. A 99, 032331 (2019).
  * Mari  _et al._ [2020] A. Mari, T. R. Bromley, and N. Killoran, Estimating the gradient and higher-order derivatives on quantum hardware, Physical Review A 103, 012405 (2020).
  * ibm [2021] 5-qubit backend: IBM Q team, “IBM Q 5 Yorktown backend specification V2.2.5,". Retrieved from https://quantum-computing.ibm.com (2021).
  * Li and Benjamin [2017] Y. Li and S. C. Benjamin, Efficient variational quantum simulator incorporating active error minimization, Physical Review X 7, 021050 (2017).
  * Endo  _et al._ [2018] S. Endo, S. C. Benjamin, and Y. Li, Practical quantum error mitigation for near-future applications, Physical Review X 8, 031027 (2018).
  * Barron and Wood [2020] G. S. Barron and C. J. Wood, Measurement error mitigation for variational quantum algorithms, arXiv preprint arXiv:2010.08520 (2020).