[1]Mikio Nakahara

1]IQM Quantum Computers, Keilaranta 19, Espoo, 02150, Finland

2]IQM Quantum Computers, Georg-Brauchle-Ring 23-25, Munich, 80992, Germany

# On-Premises Superconducting Quantum Computer for Education and Research

Jami Rönkkö  Olli Ahonen  Ville Bergholm  Alessio Calzona  Attila Geresdi
Hermanni Heimonen  Johannes Heinsoo  Vladimir Milchakov  Stefan Pogorzalek
Matthew Sarsby  Mykhailo Savytskyi  Stefan Seegerer  Fedor Šimkovic IV  P.V.
Sriluckshmy  Panu T. Vesanen<EMAIL_ADDRESS>[  [

###### Abstract

With a growing interest in quantum technology globally, there is an increasing
need for accessing relevant physical systems for education and research. In
this paper we introduce a commercially available on-site quantum computer
utilizing superconducting technology, offering insights into its fundamental
hardware and software components. We show how this system can be used in
education to teach quantum concepts and deepen understanding of quantum theory
and quantum computing. It offers learning opportunities for future talent and
contributes to technological progress. Additionally, we demonstrate its use in
research by replicating some notable recent achievements.

###### keywords:

quantum computer, transmon qubits, quantum algorithms

##  1 Introduction

Quantum computing is a promising technology which is expected to efficiently
solve certain classes of problems that are challenging for classical computers
in terms of computational time and/or hardware resources. The term _quantum
advantage_ is coined for the demonstration of this algorithmic speed-up on
quantum hardware. Several quantum algorithms have been devised to demonstrate
this, for example prime number factorization of large integers [1] with an
exponential speed-up compared to its best known classical counterpart. A
similar speed-up exists when simulating the chemical and physical properties
of molecules and the dynamics of fundamental physical models [2]. We should
note that most of these algorithms assume an error-free quantum hardware with
a number of quantum bits or qubits beyond the reach of current technology.
With the inherent presence of the loss of quantum information in any physical
system, a fault-tolerant quantum computer [3] would employ a built-in quantum
error correction, where the number of error-free logical qubits is less than
the error-prone and noisy physical qubits.

However, even in the absence of error-correction, noisy intermediate-scale
quantum (NISQ) computers [4] are thought to exhibit quantum advantage over
classical high-performance computers (HPC) in the range of 100 to 1000 qubits,
depending on the quality of the quantum hardware and the connectivity between
the qubits. Among many physical platforms, superconducting quantum hardware is
well-suited for scaling the number of qubits and improving their fidelity
while maintaining connectivity and thus becomes a preferred technology in the
NISQ era with roadmaps towards fault tolerance [5].

Here, we introduce the IQM SparkTM [6] prototype111This research paper is
based on a prototype system of a commercial IQM SparkTM and its system
specifications and performance may therefore differ from the commercial IQM
SparkTM. The authors will not make any statements relating to the commercial
IQM SparkTM or give any guarantees relating to the system specification and
performance of the commercial IQM SparkTM., a 5-qubit superconducting quantum
computer designed and developed to enable a low-barrier access to both its
hardware and software components. The hardware is self-contained with a
packaged superconducting quantum processing unit (QPU), a dilution
refrigerator, control electronics, while the software components allow for
both a direct manipulation of the qubits by microwave pulses or to run small
scale quantum algorithms composed of quantum gates [6]. As we will demonstrate
with different use cases, this system can be harnessed for a range of
educational activities from teaching the concepts of superconducting quantum
hardware to developing an understanding of quantum error mitigation and
performing experiments from different fields of research.

The rest of the paper is organized as follows. In Section 2, we introduce the
hardware, in which, after the overview, the basics of the transmon qubits,
tunable couplers, traveling wave parametric amplifiers, dilution
refrigerators, and control electronics are explained. In Section 3, we
introduce the software necessary to operate the quantum computer. The next two
sections are dedicated to applications of a quantum computer to education and
research. Some applications are available only for an on-premises quantum
computer. In Section 4 we introduce use cases for educational purposes, namely

  * •

calibration,

  * •

benchmarking

  * •

visualization of pulses with oscilloscope

  * •

error mitigation, and finally

  * •

execution of simple quantum algorithms.

In Section 5, we reproduce some research results which appeared recently in
scientific journals, namely

  * •

simulation of neutrino oscillation,

  * •

estimation of Jones polynomials, and

  * •

an introduction into embedding techniques for quantum chemistry.

Section 6 is devoted to summary and discussion.

### Notation

Throughout this paper, $I_{k}$ stands for the $k$-dimensional unit matrix.
Pauli matrices are denoted by $X,Y$ and $Z$, with an optional subscript to
denote the qubit they apply to:

$P_{i}=I_{2}\otimes\ldots\otimes I_{2}\otimes P\otimes
I_{2}\otimes\ldots\otimes I_{2},\quad P\in\\{X,Y,Z\\},$

where $P$ is in the $i$th position. $X_{1}Z_{3}=X\otimes I_{2}\otimes Z$, for
example. We use the common convention for qubit ordering, namely the top qubit
in a quantum circuit is the first qubit, which is opposite to the Qiskit
convention [7]. We assign qubit numbers 1 to 5 as shown in Fig. 1 although
Qiskit assigns 0 to 4. Which convention is employed should be obvious from the
context.

### Estimating expectation values of observables

A quantum computer estimates the probabilities of the $Z$-basis states of the
measured qubits by repeating the same multiqubit $Z$-basis measurement many
times, and computing the relative frequencies of the outcomes. The number of
repetitions is called “shots”. For example, for a single-qubit state
$|\psi\rangle=a|0\rangle+b|1\rangle$ we may estimate the probabilities
$p_{0}=|a|^{2}$ and $p_{1}=|b|^{2}$. Using these values we may estimate the
expectation value of $Z_{i}$ as $\langle\psi|Z_{i}|\psi\rangle=p_{0}-p_{1}$.
By using the identities

$X=H\,Z\,H\quad\text{and}\quad Y=S\,H\,Z\,H\,S^{\dagger},$ (1)

we may estimate the expectation value of $X$ or $Y$ by first rotating the
state $|\psi\rangle$ by $H$ or $H\,S^{\dagger}$, respectively, and then
estimate the expectation value of $Z$. For execution on our hardware, all
circuits are transpiled into single-qubit
$R(\theta,\phi)=\exp\left[-i\theta\left(\cos\phi\,X+\sin\phi\,Y\right)/2\right]$
gates and two-qubit $CZ$ gates. In this process the extra gates used for
estimating $X$\- or $Y$-expectation values are often merged with adjacent
single-qubit gates in the circuit.

##  2 Hardware

Our superconducting quantum computer is a full-stack system consisting of a
5-qubit superconducting QPU, dilution refrigerator, optimized cryogenic
microwave and DC lines, control electronics, and appropriate classical
computing hardware to run the control software. The details of our quantum
computer used in the experiments of this paper are described in the following
subsections.

###  2.1 Quantum processing unit

####  2.1.1 Overview

Figure 1: Design of a 5-qubit superconducting quantum processing unit employed
in this paper, showing 5 qubits (QB) connected by 4 tunable couplers (TC).
Black, apart from explanatory text, indicates areas where superconducting film
is etched exposing the substrate.

The core of any quantum computing system is the quantum processing unit (QPU)
comprising of qubits, qubit-couplers and control and readout lines. The QPU of
our quantum computer features five data qubits with a single central qubit
connected to the four peripheral qubit through tunable couplers [8, 9] in star
topology as depicted in Fig. 1. The qubits and tunable couplers are described
below in more detail.

The chip layout is drawn using KQCircuits [10]. It is a free open-source add-
on for KLayout [11], a widely used open-source operating system independent
layout viewer and editor for integrated circuits. KQCircuits adds the
necessary functions to KLayout to programmatically, or using graphical user
interface, draw superconducting circuits and export the designs for
fabrication or for standard microwave simulation software. KQCircuits also
exports netlists, which enable SPICE-like quasi-lumped-element simulations for
fast validation and optimization of geometrical parameters to achieve target
coupling strengths between circuit elements discussed below.

####  2.1.2 Qubit type

The states of computational qubits in our QPU are physically stored in non-
linear oscillators referred to as transmon qubits. Transmon is a modified
charge qubit, where a Josephson junction or a Superconducting QUantum
Interface Device (SQUID) is shunted by a large capacitor in a way that
Josephson energy exceeds capacitor energy by a factor of few tens [12]. The
transmon is prevalent qubit type in superconducting quantum computation due to
its stability against charge and flux noise, and simplicity of operation. In
our qubit circuit, sometimes referred to as a grounded transmon [13], the
qubit capacitor is formed by a thin metal film island separated from the
coplanar ground plane by a gap where metal has been etched and underling
dielectric is exposed. In addition to the mutual capacitance, the central
island is connected to the ground via the SQUID consisting of parallel-
connected Josephson junctions.

The shape of the qubit charge island has six-fold rotational symmetry. Each
sector features a capacitor island. Each of the islands has its own size,
which allows the coupling capacitance to be individually tuned to achieve
target coupling to the neighboring qubits, qubit state readout resonator and
extra shunt to the ground for precise targeting of total shunt capacitance. In
between two coupling islands, there is a narrow strip of charge island to
reduce coupling between the neighboring couplers.

####  2.1.3 Qubit control

Each of the qubits is individually addressed by two control lines. Control
lines are implemented as coplanar waveguides.

The center conductor of the flux line is shorted to the ground in the vicinity
of the qubit SQUID creating an effective mutual inductance between the center
conductor and the SQUID loop. By applying electrical current through the flux
line, magnetic flux created through the SQUID loop creates a phase bias across
the Josephson junctions, reducing the effective Josephson energy of the SQUID
and hence the qubit frequency. At the maximum qubit frequency, the frequency
is insensitive, to the first order, to external flux maximizing the coherence
time of the qubit and is referred to as a sweetspot. Qubit frequency changes
are used to find overall optimal operation frequencies, change dispersive
coupling rates to the other elements, and implement physical $Z$ and $CZ$
gates, more on two-qubit gates below.

For drive lines, the center conductor is left open circuited and due to
proximity has mutual capacitance to the charge island of the qubit. By
applying microwave signals through the drive line, the qubit state can be
driven between internal states with the energy difference corresponding the
signal frequency. The capacitance value is carefully chosen to ensure
sufficiently low qubit coupling to the $50\,\Omega$ environment to minimize
Purcell losses [14] whilst keeping the coupling to the target qubit strong
compared to the spectator qubits. Resonant qubit drive is used to implement
$X$, $Y$ or any $R(\theta,\phi)$ gate. By choosing corresponding drive
frequency, gates between higher energy levels of the qubit can also be
achieved giving access to the Hilbert space of larger dimension.

####  2.1.4 Readout

To infer the state of a superconducting qubit, so-called dispersive readout is
employed [15, 16]. This is a widely used method which employs transverse
coupling between the qubit and resonator based on a dipole-dipole interaction.
Due to the transverse coupling term in the Jaynes-Cummings Hamiltonian [17], a
qubit-state-dependent frequency shift of the resonator, known as the
dispersive shift, is observed. Every qubit has a dedicated readout resonator
connected to it and each readout resonator has a different resonance
frequency.

To suppress the Purcell decay rate of the qubits through the readout
resonators, the resonator is not directly coupled to the
$50\text{\,}\mathrm{\SIUnitSymbolOhm}$ environment. Instead, each readout
resonator couples to an individual Purcell filter \- a bandpass filter which
reduces the transmission at the qubit frequency.

As shown in Fig. 2, the Purcell filters are in turn all coupled to a common
probe line. The state of the qubit registry is inferred by probing the
transmission of the probe line with a frequency comb and comparing the phase
and amplitude of the transmitted signal at the frequency of each readout
resonator individually to a set threshold. The readout cross-talk is reduced
thanks to the individual Purcell filters [18]. The amount of coupling and the
frequency detuning between the elements is optimised to balance readout speed
and Purcell relaxation rate. The input capacitor and the shunt at the output
of the probe line forms another resonator, where the total length defines the
frequency and the location of the output tap defines coupling strength to the
output port, see the right side of Fig. 2.

Figure 2: Quasi lumped element circuit diagram of readout circuit, including
readout- and Purcell resonators connected to probe line, which consists of a
distributed Purcell filter. Qubits are depicted as circles with two horizontal
lines.

The coupling strengths and detunings are chosen such that the Purcell effect
would not limit the intrinsic $T_{1}$.

####  2.1.5 Tunable couplers

Tunable couplers are utilized in order to perform two-qubit gates between the
above-mentioned qubits. Tunable couplers are circuit components based on
transmon qubits, which enable us to perform two-qubits gates with state of the
art fidelities above 99% [9]. The main benefit of using tunable couplers is
the possibility to compensate the native $ZZ$-interaction between qubits which
enables high fidelity identity gates [8]. In our design, see Fig. 3, the
interaction between tunable coupler and qubits is mediated by waveguide
extenders [9]. This feature allows us to place a significant distance between
the qubits to avoid inter-qubit cross-talk, while keeping the switchable
$ZZ$-coupling large enough and, in larger devices, fit the readout resonators
into the qubit lattice unit cell.

Figure 3: A quasi-lumped element circuit diagram of two transmon qubits (blue
and orange) coupled by a tunable coupling structure consisting of waveguide
extenders (turquoise) and a floating coupler qubit (red) [9]. Electrical nodes
are marked with capital letters. Grey color elements represent the effective
couplings implemented by the waveguide extenders.

By applying an external magnetic flux into the SQUID loop of a tunable
coupler, one can change a coupler frequency and thus the effective amount of
the $ZZ$-interaction $g_{zz}$ between relevant pair of qubits. The effective
value of the $g_{zz}$ can be changed in a wide range including both positive
and negative values. Consequently, there exists a point where this interaction
is equal to zero. This point is used while the QPU is idling so that all qubit
pairs have negligible interaction and all the native couplings between pairs
of qubits are compensated. To perform a two-qubit gate between neighboring
qubits, the coupler can be flux-tuned by a square-like baseband pulse to
change its frequency causing an interaction between qubits for a certain
amount of time, see also subsection 2.5.

By choosing the detuning between qubits during a gate operation, one can
implement $CZ$, $i$SWAP gates or any general fermionic simulation gate [19]
with the same hardware.

###  2.2 QPU Package

The QPU is mounted inside a carrier for handling, shielding, mounting, and
signal connection purposes. The shape of the carrier is optimized to make all
standing wave modes be far-detuned from operational frequencies. The QPU is
wire-bonded to a printed circuit board (PCB) with coplanar waveguides that is
also attached to the carrier. The PCB serves to transmit signals between QPU
launchpads and external microwave connections. The carrier is manufactured out
of high conductivity copper for improved thermalisation and reducing potential
interactions with impurities in the metal. This carrier has a gold plated
finish for better thermal contact between mating surfaces, to reduce losses of
signals in the exposed transmission lines, and to ensure high quality factors
of all mentioned standing wave modes. The particular gold plating process used
is non-magnetic to maintain a clean magnetic field environment. The sample
carrier securely holds the QPU, protects the chip from stray radiation,
minimizes microwave interference and cross-talk between the signals.

To thermally attach the QPU carrier to the refrigerator, it is mounted to a
copper cold finger, which is situated inside a multiple layered magnetic
shielding assembly. These shields are required for minimizing the interaction
with external magnetic fields and suppress to environmental radiation.

###  2.3 Refrigerator

The cold finger with the chip carrier is attached to the mixing chamber plate
of a commercial BlueforsTM refrigerator in order to cool the QPU to the
operating temperature of a few tens of mK. These machines use a pulse tube
refrigerator to cool the first stages to a few Kelvin and then use a dilution
refrigerator to reach the base temperature. The experiments presented in this
paper were performed with components attached to the experimental stage with a
temperature of 30 mK or cooler.

###  2.4 Signal inputs and outputs

The microwave signals for the qubit drive, tunable coupler flux, parametric
amplifier pump, and readout probe are routed from room temperature to the QPU
by coaxial SCuNi wires that include appropriate attenuation cascade at the
different temperature stages as well as low-pass filtering at the base
temperature. DC signals for qubit flux are routed using twisted pair wiring
with low-pass filtering at room temperature, the 3 K stage, and at the base
temperature stage.

The readout response is amplified by a Traveling Wave Parametric Amplifier
(TWPA), routed via appropriate isolation upwards by superconducting coaxial
NbTi wiring to a High-Electron-Mobility Transistor (HEMT) amplifier operating
at a nominal 3 K temperature. After the HEMT the signal is carried by silver
plated copper-nickel coaxial wiring to reach the top plate of the cryostat.

####  2.4.1 Traveling wave parametric amplifier

Qubit readout relies on readout pulses with as little energy as tens of
microwave photons. To detect such weak signals, quantum limited amplifiers are
used, where amount of added noise is limited by quantum mechanics [20]. The
first amplifier in our readout chain is a TWPA. Due to high gain and
bandwidth, it enables frequency multiplxed readout of all qubits within
$100\text{\,}\mathrm{ns}$ and readout fidelity limited only by qubit decay
time. [21]

Parametric amplifiers include nonlinear media, where propagating weak and
strong tones exchange energy. In TWPAs, the nonlinear media consists of long
series of Josephson junctions [22] forming an analogue of an optical Kerr
medium. The Josephson potential being an even function of the superconducting
phase difference $\phi$, the provided nonlinearity to the lowest order is of
the form $\phi^{4}$ which enables four-wave mixing (4WM). Here two photons
from a strong pump tone generate one photon in phase with the weak input
signal and another idler photon [23]. Pump, signal and idler tone frequencies
are related by the conservation of energy [20], see Fig. 4.

Figure 4: Energy level diagrams for (a) 4WM and (b) 3WM processes.
$\omega_{p}$, $\omega_{s}$ and $\omega_{i}$ represent the angular frequencies
of pump, signal and idler photons, respectively. $\Delta\omega$ represents an
frequency detuning from the degenerate mode of amplification.

In the presence of an external flux, the lowest order of non-linearity becomes
of the form $\phi^{3}$. This term can facilitate a three-wave mixing (3WM)
process where a single pump photon at roughly twice the signal frequency gives
its energy to a pair of signal and idler photons [24, 25, 26]. Having the pump
tone far from the signal frequency is beneficial, as then the strong tone at
the output of the TWPA can be removed with a simple filter and one avoids any
compression effects in later amplification stages.

If the signal and idler frequencies are the same, the resulting amplification
is said to be degenerate and only one of the signal quadratures is amplified,
but possibly without any added noise. In our system, we typically employ 3WM
TWPAs in the non-degenerate regime such that we can frequency multiplex the
readout. However, by changing the flux bias current it is possible to tune the
degeneracy to a frequency of interest without changes to the hardware.

###  2.5 QPU control electronics

The microwave drive pulses are generated by conventional AC-coupled microwave
arbitrary waveform generators (AWGs) operating at the qubit frequency band.
The tunable coupler flux pulses are generated by a DC-coupled baseband AWG.
The readout probe signals are generated and acquired by a conventional quantum
analyzer operating at the frequency band of the readout resonators. The
readout and drive instruments also provide a combined functionality that
enables fast feedback, i.e., driving signals dependent on the readout result
at time scales shorter than the qubit coherence times. The qubit flux and TWPA
bias currents are generated by a DC voltage source which is connected to the
QPU and TWPA devices via a cascade of low pass filters that include inline
resistance to convert voltage to a stable direct current.

The electronics racks of the system include all the required auxiliary
electronics for operating a full-stack quantum computer: uninterruptible power
supply (UPS) to provide regulation and filtering of the mains power to the
measurement electronics rack, a network remote configurable mains power
distribution unit (ePDU), power supplies for the various readout amplifiers, a
reference clock, power supplies for the DC sources, a main Linux host for
running the control software, a Windows host for running the Bluefors software
that controls the dilution refrigerator, several smaller specialised Linux
hosts for instrumentation interfaces, a dedicated firewall, and a network
switch that provides Ethernet connectivity to the hosting facility.

##  3 Software

Figure 5: The software layers and modules of our quantum computer control
software stack.

The software stack of our quantum computer is divided into different
functional layers presented in Fig. 5. It can be interfaced in several ways
based on the required level of access. The modules and interactions of the
software stack are described in more details in the following subsections.

###  3.1 Cortex

Cortex is a set of software components for running quantum algorithms on our
quantum computer. It is the highest level of abstraction in the control
software stack of our on-premises quantum computers. Cortex focuses on
enabling computation for the end user, rather than experimenting with the
behaviour of the individual elements of the quantum computer.

Cortex allows users to define and execute quantum algorithms on the quantum
computer, expressed as quantum circuits using high-level frameworks and
description languages such as Cirq, Qiskit, and OpenQASM 2.0. The input to the
server is a computation job containing one or more quantum circuits to be
executed, the number of shots, and possibly some other parameters. The job is
queued for execution on the quantum computer. The results of the measurements
of the circuits are returned when the job is completed.

###  3.2 EXA

EXA is a Python-based framework for characterising, calibrating, and
controlling our quantum computer. EXA supports the execution of pre-defined
experiments as well as the definition and execution of new custom experiments.
An experiment unit combines different functionalities such as execution flow,
data manipulation, analysis, and presentation, and it can also be built in a
modular way using other experiments.

Using the EXA experiment library, users can create Jupyter notebooks or
standalone Python applications e.g. to implement macro-like capabilities to
simplify the control and measurement processes, eliminate standard repetitive
operations using automated procedures, and develop entirely new experiments.

###  3.3 IQM Station Control

IQM Station Control takes care of low-level functionality such as housing
instrument parameters and hardware drivers. It hides low-level hardware
details from the higher-level components, EXA and Cortex. Both Cortex and EXA
communicate with Station Control service via its non-RESTful JSON (JavaScript
Object Notation) HTTP (Hypertext Transfer Protocol) interface. The interface
provides endpoints for performing various parameter sweeps and executing pulse
schedules. In normal use, the user does not need to interact directly with the
service.

Station Control uses device-specific drivers to further encapsulate the
details of each instrument, including its low-level communication protocol.

##  4 Applications to education

A small scale on-premises quantum computer is exceptionally useful for
educational purposes. It facilitates hands-on experimentation, allowing
students not only to run quantum circuits, but also to conduct pulse-level
experiments, change the calibration, or connect external periphery. In
general, applications to education can be sorted into two categories, (1)
experiments/lab sessions that involve accessing the hardware physically or
through the pulse-level interface, and (2) accessing the quantum computer
through the circuit-level interface.

###  4.1 Utilizing hardware access/pulse-level access for education

Lab sessions that involve accessing the hardware physically or utilizing the
pulse-level interface and changing the configuration or calibration of the
device help engage students and provide additional learning opportunities that
are pivotal in cultivating the next generation of quantum scientists and
engineers. Below we will illustrate four examples of how an on-premises
quantum computer can be used for this purpose in education.

####  4.1.1 Exploring a quantum computer

Access to a physical quantum computer enables the investigation of the setup
of these machines. Together with the appropriate exercises and learning
materials this kind of physical access bridges the gap between abstract
quantum circuit descriptions and the actual superconducting quantum computer
that executes them. Students experience first hand how the qubits are
protected from the environment via cooling and magnetic shields, and also how
information is exchanged with the classical world.

####  4.1.2 Calibrating a quantum computer

By utilizing the pulse-level interface, learners can explore the importance of
calibration and create their own calibration sets. Calibration is required to
find the optimal parameters to operate the QPU and is crucial for high
fidelity operations. In a lab setting, the learners create and apply their
calibration sets and compare outcomes to understand the impact of calibration
on the results as shown in Fig. 6 (a). Figure 6 (b) shows the measurement
outcomes of the 5-qubit GHZ state

$|\Psi\rangle=\frac{1}{\sqrt{2}}(|00000\rangle+|11111\rangle)$ (2)

with two different calibration sets.

Furthermore, learners can investigate the fidelity of gates and the potential
causes of discrepancies in calibration outcomes, encouraging students to
critically analyze their results against provided benchmarks. Benchmarking can
also expand to aspects such as $T_{1}$ and $T_{2}$ times or different
strategies for assessing performance.

Figure 6: (a) Learners may compare their calibration with another. They notice
the badly calibrated $CZ$ gate in the second calibration set, indicated by the
red arrow, and investigate the cause. (b) Comparison of the 5-qubit GHZ state
preparation fidelity using different calibration sets shown in (a).

####  4.1.3 Exploring control waveforms

With physical access to the device, students can also plug in selected
peripheral devices such as oscilloscopes to further investigate the connection
of software-defined operations and the physical implementation of quantum
operations. Multiple smaller experiments guide the learners through
investigating control pulse characteristics and qubit manipulation. For
example, they execute multiple instructions that rotate the qubit state by
different angles and measure the corresponding pulse shapes. As a more
complicated example, learners can explore the control pulse schedules that
result from multi-qubit circuits such as one that generates a Bell state
$|\Phi_{+}\rangle\,{=}\,(|00\rangle+|11\rangle)/\sqrt{2}$, as shown in Fig. 7
(a). The resulting control and readout pulses measured with an oscilloscope
are depicted in Fig. 7 (b). Through this hands-on approach, learners will gain
insight into the control electronics that enable superconducting quantum
computing and the concepts of pulse control.

Figure 7: (a) Transpiled circuit that creates a Bell state using native
operations of our quantum computer. Note that this circuit has been chosen for
illustrative purposes and has not been fully optimized for the given
architecture. (b) Resulting pulse shapes as displayed on the oscilloscope
interface. Horizontal and vertical axis depict time and voltage, respectively.
Curves are shifted vertically for better visibility.

####  4.1.4 Multi-level quantum hardware

Direct hardware access allows one to investigate physical quantum systems
beyond the two-level approximation defining a qubit. Here we demonstrate the
state preparation and readout of the second excited state of a transmon. This
example provides interested learners with a better understanding of the actual
superconducting quantum hardware. Furthermore, it connects the educational
value of the system with recent scientific results enabled by utilizing the
multi-level nature of the transmon [27], such as three-level (qutrit) quantum
processors [28], tunable coupler architectures using the second excited state
[9] and fast single-qubit gates by the shortcuts-to-adiabaticity version of
the stimulated Raman processes (STIRAP) [29].

Figure 8: State preparation and relaxation dynamics of a qutrit. The inset
displays the pulse sequence, which starts with preparing the qutrit in the
second excited state, and the readout is performed after a variable delay
time. The result of the single shot analysis is shown as dots, and the best
fit to the Markovian model in the inset is displayed as solid curves with the
relaxation rates $\Gamma_{21}$, $\Gamma_{20}$ and $\Gamma_{10}$ given in the
text.

In Fig. 8, we display the results of an experiment addressing the relaxation
dynamics of a transmon prepared in the $|2\rangle$ state by a calibrated
$\pi_{0-2}$ pulse. The population then freely evolves and state discrimination
is performed after a delay time (see inset). The observed evolution (dots) of
the three states $|0\rangle$, $|1\rangle$ and $|2\rangle$ is then fitted
(solid lines) to extract the relaxation timescales. We find
$\Gamma^{-1}_{10}=44.4\pm 1.3\,\mu$s, $\Gamma^{-1}_{21}=35.0\pm 1.1\,\mu$s and
$\Gamma^{-1}_{20}=69.2\pm 4.1\,\mu$s, which together capture the expected
dynamics of the qutrit, where the inequalities
$\Gamma^{-1}_{10}>\Gamma^{-1}_{21}$ and
$\Gamma^{-1}_{10},\Gamma^{-1}_{21}<\Gamma^{-1}_{20}$ demonstrate the role of
the transition matrix elements when describing the relaxation of a quantum
system [30]. We conclude that pulse-level access enables the direct
investigation and control of the superconducting quantum hardware beyond what
higher abstraction layers can provide, making it a vital tool for educational
programs targeting quantum hardware.

###  4.2 Utilizing circuit-level access for education

A lot of interest goes into investigating, creating and improving NISQ
algorithms. This subsection will demonstrate the use of gate-based access in
educational settings by providing different examples that can be practiced by
learners. The output of each quantum circuit is illustrated by executing it
with our superconducting quantum computer.

Because of the unavoidable presence of errors during algorithm execution on
NISQ devices such as our quantum computer, the results are expected to differ
from the ideal, noiseless ones. Importantly, there are strategies, often
collectively referred to as “quantum error mitigation”, that can be employed,
either individually or in combination with each other, to reduce the effects
of errors on the execution of a desired algorithm [31]. In particular, in the
examples that will follow, we make use of techniques belonging to three
different classes.

  * •

“Error suppression” techniques aim to modify and reduce the effects of errors
at the level of each single circuit run. A prominent example is the randomized
compiling (RC) technique [32, 33] that, through random Pauli twirling,
effectively converts problematic coherent errors (e.g. systematic
overrotations associated with a given gate) to stochastic errors, which add up
more favourably and whose effects are easier to mitigate further.

  * •

“Readout error mitigation” (REM) techniques target errors that occur during
the measurement of the qubits [34, 31]. In general, they consists of two
steps. At first, a set of simple and shallow characterization circuits is
executed on the hardware, in order to determine the properties and the
magnitude of the readout errors. As a simple example, one might want to
measure what is the probability that a given qubit prepared, say, in the
$|0\rangle$ state is incorrectly measured to be in $|1\rangle$. Once this
information is gathered, the desired algorithm is executed and its raw results
are post-processed in order to compensate for the readout errors. In this
paper, we have mitigated readout errors using correlated readout error
mitigation calibrated with 10,000 shots per basis state.

  * •

A third class of “error mitigation” techniques mainly target errors happening
at the gate level, i.e. during the execution of the bulk of the circuits. This
is generally achieved by executing different variants of the desired quantum
circuit and by combining their output via classical post-processing, leading
to a (potentially significant) run time overhead [31]. Despite this trade-off
between quality and speed, the implementation of error mitigation strategies
is a key element for successful algorithm execution in the NISQ era. One
simple-yet-effective technique to mitigate gate errors is known as zero noise
extrapolation (ZNE) [35], which is based on the idea of artificially
increasing the noise level and then making use of noisier results to
extrapolate back to the noiseless limit. Open source and educational
implementations of several other techniques can be found, for example, in
[36].

####  4.2.1 Violation of the CHSH inequality

The Nobel Prize in Physics 2022 was awarded jointly to Alain Aspect, John F.
Clauser and Anton Zeilinger “for experiments with entangled photons,
establishing the violation of Bell inequalities and pioneering quantum
information science” [37]. Bell’s theorem claims that correlations of
measurement outcomes of two experimenters separated from one another have an
upper bound if nature follows the principle of local realism [38]. Suppose two
parties, often called Alice and Bob, are located far away from each other in
the experiment. They both have two observables they can measure of some signal
that comes to them, but they cannot measure both simultaneously. Bell-type
inequalities define correlators over the possible measurement settings and
their outcomes, assuming that the measurements happen so fast that no
communication is possible between the measurement devices of Alice and Bob due
to the finite speed of light [39]. A Bell inequality then separates so-called
local probability distributions from non-local distributions, that have
stronger correlations than the local distributions. The most famous Bell-type
inequality is the CHSH inequality [40], which is straightforward to test with
two qubits on a quantum computer. The measurements of the qubits are
definitely not space-like separated events, though. The inequality is given by
the formula:

$-2\leq E(QS)+E(RS)+E(RT)-E(QT)\leq 2,$ (3)

if the local realism is true. Here the observables are

$Q=X_{1}\quad R=Z_{1}\quad S=Z_{2}\quad T=Z_{2}$

where index $1$ refers to Alice’s and $2$ to Bob’s qubit and $E(A)$ stands for
the expectation value of $A$.

Let us now consider how these correlations behave in quantum mechanics, and
evaluate the above expectation values in the state

$|\Psi(\theta)\rangle=R_{y}(\theta)_{1}(|00\rangle+|11\rangle)/\sqrt{2},$ (4)

where $R_{y}(\theta)=\exp(-i\theta Y/2)$. Calculating the expectation values
as described at the end of section 1, we obtain

$E(QS)=E(RT)=\cos\theta,\quad E(QT)=-E(RS)=\sin\theta,$ (5)

and thus

$E(QS)+E(RS)+E(RT)-E(QT)=2\sqrt{2}\cos(\theta+\pi/4).$ (6)

Clearly the CHSH inequality in Eq. (3) is violated in the state
$|\Psi(\theta)\rangle$ iff

$\theta\in\left(\frac{\pi}{2},\pi\right)\sqcup\left(\frac{3\pi}{2},2\pi\right),$

which demonstrates that quantum mechanics is not compatible with local
realism.

To experimentally test this prediction, we will execute a parameterised
quantum circuit given in Fig. 9 (a) to create the state $|\Psi(\theta)\rangle$
and measure the relevant expectation values for multiple values of $\theta$.
Figure 9 (b) shows how the CHSH observable oscillates as we rotate the state,
and violates the equality as predicted. The statistical uncertainty due to the
finite amount of shots is shown by the error bars, which correspond to one
standard deviation. They have been determined by means of bootstrapping, i.e.
by classically resampling several times the probability distribution obtained
from each circuit run, thus obtaining a set of reprocessed results that can be
used to estimate confidence intervals. We use this method throughout the
paper.

(a)

(b)

Figure 9: (a) Quantum circuit that prepares $|\Psi(\theta)\rangle$. (b)
Expectation value of the CHSH observable Eq. (6) in the state
$|\Psi(\theta)\rangle$, plotted over the $y$ rotation angle $\theta$. The raw
experimental data points are shown with blue dots; the implementation of
readout error mitigation (green crosses) brings them closer to the ideal
noiseless result (red curve). There are two regions outside the black
horizontal lines, where the CHSH inequality is violated and hence non-locality
is demonstrated. The statistical uncertainty is so small that the error bars
remain within the markers.

####  4.2.2 5-qubit GHZ state, decoherence and Mermin’s inequality

5-qubit GHZ state

Let us prepare a maximally entangled 5-qubit state and see what we can do with
it. The 5-qubit GHZ state [41]

$|\Psi_{5}\rangle=\frac{1}{\sqrt{2}}(|00000\rangle+|11111\rangle)$ (7)

is one of the maximally entangled 5-qubit states. This state is obtained by
applying the quantum circuit in Fig. 10 (a) on $|00000\rangle$. Figure 10 (b)
shows the output histogram of our quantum computer obtained with 5,000 shots,
while Fig. 10 (c) shows the output after readout error mitigation is applied.

Figure 10: (a) Quantum circuit to implement the 5-qubit GHZ state. (b) Output
of 5-qubit quantum computer. (c) Output after readout error mitigation is
applied.

Entanglement, mixed state and decoherence

Let us separate the 5-qubit system into subsystems made of qubits 12 and
qubits 345. Suppose one measures an observable $O$ associated with subsystem
12. The expectation value of $O$ with respect to $|\Psi\rangle$ is

$\langle O\rangle=\langle\Psi|(O\otimes I_{8})|\Psi\rangle=\frac{1}{2}\langle
00|O|00\rangle+\frac{1}{2}\langle 11|O|11\rangle,$ (8)

which is an expectation value with respect to a mixed state, even though the
total system is in a pure state. This is directly demonstrated by evaluating
the density matrices of (a) the 5-qubit GHZ state, (b) the subsystem 12 and
(c) the subsystem 345 as

$\rho_{\mathrm{GHZ}}=\frac{1}{2}\begin{pmatrix}1&0&\ldots&0&1\\\
0&0&\ldots&0&0\\\ \vdots&&\vdots&&\vdots\\\ 0&0&\ldots&0&0\\\
1&0&\ldots&0&1\end{pmatrix},$ (9) $\displaystyle\rho_{\mathrm{12}}$
$\displaystyle=$ $\displaystyle\sum_{i,j,k\in\\{0,1\\}}(I_{2}\otimes
I_{2}\otimes\langle ijk|)\rho_{\mathrm{GHZ}}(I_{2}\otimes
I_{2}\otimes|ijk\rangle)=\frac{1}{2}\mathrm{diag}(1,0,0,1)$ (10)
$\displaystyle\rho_{\mathrm{345}}$ $\displaystyle=$
$\displaystyle\sum_{i,j\in\\{0,1\\}}(\langle ij|\otimes I_{2}\otimes
I_{2}\otimes I_{2})\rho_{\mathrm{GHZ}}(|ij\rangle\otimes I_{2}\otimes
I_{2}\otimes I_{2})$ (11) $\displaystyle=$
$\displaystyle\frac{1}{2}\mathrm{diag}(1,0,0,0,0,0,0,1),$

respectively. Figure 11 shows the result of quantum state tomography for (a)
the GHZ state (b) subsystem 12 and (c) subsystem 345, which are obtained
experimentally with the Qiskit state tomography algorithm. The subsystem 12 is
in a mixed state since it cannot access the information of the subsystem 345
and vice versa.

(a)

(b)

(c)

Figure 11: State tomography of (a) 5-qubit GHZ state, (b) qubits 12 (principal
system) and (c) qubits 345 (environment). Theoretically (a) is a pure state
with rank 1 while (b) and (c) are mixed states with rank 2. We used local
readout error mitigation and 3500 shots per measurement basis for the
tomography.

This situation models decoherence. It is often said that interaction of a
quantum system with environment causes entanglement between the two systems,
by which a initial pure state of the system becomes a mixed state. This
explanation of decoherence is often difficult to understand for beginners. Let
us call the subsystem 12 the principal system while subsystem 345 the
environment. The total system, started in a pure tensor product state
$|00\rangle\otimes|000\rangle$, evolves to the entangled GHZ state under the
unitary time evolution given by Fig. 10 (a). The total system is still in a
pure state. Although the principal system was in a pure state $|00\rangle$ in
the beginning, it is now entangled with the environment, and thus in a mixed
state $\rho_{12}$ if the environment is ignored (i.e. traced out). In other
words, the GHZ state is a purification of $\rho_{12}$.

It is interesting to evaluate the von Neumann entropy
$S(\rho)=-\mathrm{tr}\,\rho\log_{2}\rho$ of these states. Table 1 shows both
the theoretical predictions and the experimental results along with the upper
bound saturated by the uniformly mixed state. Observe that the entropy, which
is called the entanglement entropy in this context, is theoretically the same
for $\rho_{12}$ and $\rho_{345}$. Usually, entropy is proportional to the
system size but entanglement entropies derived from a pure state
$\rho_{\mathrm{GHZ}}$ are identical even though the subsystem sizes are
different.

| Theory | Experiment | Upper bound
---|---|---|---
GHZ | 0 | 0.925 | 5
$\rho_{12}$ | 1 | 1.306 | 2
$\rho_{345}$ | 1 | 1.386 | 3
Table 1: Theoretical and experimental values of the von Neumann entropy of the
5-qubit GHZ state, the subsystem 12 and the subsystem 345. Experimental data
in Fig. 11 has been employed. The right column shows the upper bound of
entropy, which is saturated by the uniformly mixed state.

Violation of Mermin’s inequality

Let us show next that Mermin’s inequality is violated by the 5-qubit GHZ
state. Mermin’s inequality is regarded as a generalization of the CHSH
inequality to multi-qubit systems [42]. The Mermin polynomial for a 5-qubit
system is defined as [43]

$\displaystyle M_{5}$ $\displaystyle=$ $\displaystyle
X_{1}X_{2}X_{3}X_{4}X_{5}$ (12)
$\displaystyle-(Y_{1}Y_{2}X_{3}X_{4}X_{5}+9~{}\mbox{permutations})$
$\displaystyle+(Y_{1}Y_{2}Y_{3}Y_{4}X_{5}+4~{}\mbox{permutations}).$

It is known that Mermin’s inequality $E(M_{5})\leq 4$ is satisfied if the
local realism holds. On the other hand, quantum theory predicts $E(M_{5})\leq
4^{2}=16$, where the upper bound is saturated if the state is maximally
entangled.

Let us evaluate $E(M_{5})$ in the 5-qubit GHZ state. Since the GHZ state is
symmetric with respect to permutations of qubits we only need to evaluate
three monomials:

$\displaystyle\langle\Psi|X_{1}X_{2}X_{3}X_{4}X_{5}|\Psi\rangle=1,$
$\displaystyle\langle\Psi|Y_{1}Y_{2}X_{3}X_{4}X_{5}|\Psi\rangle=-1,$ (13)
$\displaystyle\langle\Psi|Y_{1}Y_{2}Y_{3}Y_{4}X_{5}|\Psi\rangle=1,$

theoretically. Then we find

$\displaystyle\langle\Psi|M_{5}|\Psi\rangle=1-10\times(-1)+5\times 1=16,$ (14)

and thus the GHZ state $|\Psi\rangle$ saturates the upper bound. Let us now
confirm this prediction with our 5-qubit quantum computer.

The monomials of the Mermin polynomial can be measured as expectation values
as described in section 1. Since the qubits and gate fidelities of a NISQ
quantum computer are not homogeneous, we must measure all 16 monomials
separately. The estimation of $\langle\Psi|M_{5}|\Psi\rangle$ obtained with
our quantum computer is presented in Table 2, and clearly rules out local
realism, in favour of quantum theory.

Observable | Estimate without REM | Estimate with REM | Theoretical
---|---|---|---
$X_{1}X_{2}X_{3}X_{4}X_{5}$ | $0.5538$ | $0.9606$ | $1$
$Y_{1}Y_{2}X_{3}X_{4}X_{5}$ | $-0.4955$ | $-0.8624$ | $-1$
$Y_{1}Y_{2}Y_{3}Y_{4}X_{5}$ | $0.4398$ | $0.7689$ | $1$
$M_{5}$ | $7.708$ | $13.43$ | $16$
Table 2:  Estimated value of the Mermin polynomial in the prepared 5-qubit GHZ
state. The 16 terms of the polynomial were estimated using 10,000 shots each.
The three topmost rows in the table represent the average of all the
permutations of the Pauli operators in the given observable.

###  4.3 Maxcut Problem

Many well known quantum algorithms, such as Shor’s and Grover’s algorithms,
require a large number of qubits and fault-tolerant error correction for
useful quantum computation. This places their practical execution beyond the
capabilities of currently available NISQ computers. In contrast, variational
quantum algorithms are more suited for the NISQ computer at our hand, in that
they can be executed with a currently available number of qubits without
quantum error correction.

Variational quantum algorithms involve an optimization process, where
classical computer seeks for the optimal parameters of a quantum circuit so
that the expectation value of a Hamiltonian, representing the cost function,
evaluated with a quantum computer using the resulting state is minimized. The
parameters in the circuit are iterated many times until the expectation value
hits the minimum. Variational algorithms have many use cases e.g. in
mathematics, chemistry, finance and industrial optimizations. We will
introduce application of a variational algorithm, called QAOA (Quantum
Approximate Optimization Algorithm), to a combinatorial problem called the
Maxcut problem in this subsection. Another variational algorithm VQE
(Variational Quantum Eigensolver) will be introduced in section 5.3.

Suppose there is a graph $G$ with $n$ nodes. There are edges between some
pairs of nodes. In the Maxcut problem, one aims to partition the nodes of $G$
into a disjoint union $A\sqcup B$ such that the number of edges connecting
nodes in $A$ and $B$ is maximized.

To solve this problem, we introduce an Ising Hamiltonian

$H=\sum_{i<j}J_{ij}Z_{i}Z_{j},$ (15)

where $i$ and $j$ denote the nodes. The coupling strength is $J_{ij}=1$ if
there is an edge between nodes $i$ and $j$, while $J_{ij}=0$ if there is no
edge. Suppose $|0\rangle$ is assigned to nodes in group $A$, while $|1\rangle$
to nodes in $B$. If nodes $i$ and $j$ belong to different groups, the edge
$ij$ contributes $-1$ to the Hamiltonian, while if they belong to the same
group, the edge contributes $+1$. There is no contribution if there is no edge
connecting nodes $i$ and $j$. Thus maximizing the number of edges connecting
nodes in different groups reduces to minimizing the expectation value of the
Hamiltonian $H$ acting on an $n$-qubit system.

In fact, the above problem may be implemented with $(n-1)$-qubit system.
Suppose we find a solution $A$ and $B$ of a Maxcut problem of a given graph.
Then interchange of $A$ and $B$ is also a solution of the same problem.
Accordingly we are free to assign $|1\rangle$ to the $n$th qubit, for example,
without loss of generality. With this choise, $J_{in}Z_{i}Z_{n}$ becomes
$-J_{in}Z_{i}$. The modified Hamiltonian

$H^{\prime}=\sum_{1\leq i<j\leq
n-1}J_{ij}Z_{i}Z_{j}-\sum_{i=1}^{n-1}J_{in}Z_{i}$ (16)

is implemented with an $(n-1)$-qubit system.

Figure 12: Maxcut problem for six nodes. Node 6 is a fictitious qubit in
$|1\rangle$ state while rest are physical qubits. Solid lines represent
physical couplings while dashed lines fictitious couplings.

Let us consider a graph in Fig. 12 with 6 nodes for definiteness, where the
node 6 is a virtual node in $|1\rangle$ state while the rest are physical. By
fixing the state of node 6 to $|1\rangle$, the relevant Hilbert space is
$\mathrm{Span}(\\{|i_{1}\,i_{2}\,i_{3}\,i_{4}\,i_{5}\rangle|1\rangle\\})$,
which may be implemented with our 5-qubit quantum computer.

Figure 13 shows the solution of the Maxcut problem experimentally obtained for
the graph Fig. 12. The solution is read from the most probable state of the
probability distribution. We used single-layered QAOA ansatz and $10,000$
measurements per optimization step and readout error mitigation. Using only a
single layer of the QAOA ansatz means the algorithm has only a small number of
gates and can be executed in a short time. This implies it is an approximate
algorithm, and while it will output the correct solution, there will also be
sizable probabilities of wrong states in the distribution. The algorithm can
be more accurate and precise by increasing the layer depth, but then we start
to accumulate more errors during the execution of the algorithm, which again
introduces erroneous solutions to the output distribution of the quantum
computer. The optimal depth depends in general on the problem instances and
the error-levels of the quantum computer, and is a key implementation detail
relevant for the performance of quantum algorithms on practical problems.

Q-score

Solving the Maxcut problem has been recently adopted as a benchmark for the
practical capabilities of a quantum computer [44]. The Q-score of a quantum
computer equals the size of the graphs, whose Maxcut problem can be
sufficiently solved. The obtained cost, i.e. the average number of cut edges,
of the solution has to be above certain threshold. Specifically, one has to
find the cost of a graph to be above 0.2 on a scale where 0 corresponds to
random solution and 1 to ideal solution. The graphs chosen for the benchmark
are random Erdös-Rény graphs with 50% edge-probability between nodes.

We present a comprehensive Q-score benchmark on our 5-qubit quantum computer.
By employing the virtual node technique, we can solve the $n$-node graph with
$n-1$ qubits. Figure 14 displays Q-score results up to five physical qubits.

(a)  (b)

Figure 13: (a) Measurement outcomes of our 5-qubit quantum computer executing
QAOA for the Maxcut problem. (b) The solution of the Maxcut problem; we
achieve eight cuts, which is the maximum for this graph. The solution
$|00011\rangle$ has divided the nodes to groups 0 (green) and 1 (red), such
that the number of edges connecting nodes from different groups is maximized.
Solid lines show edges connecting nodes in the same group, while eight dashed
lines are edges connecting nodes in different groups. Figure 14: Q-score
approximation ratios $\beta(n)$ for our 5-qubit quantum computer. Ratios above
the threshold 0.2 pass the Q-score benchmark. Results of noiseless simulator
are shown to highlight the approximate nature of QAOA; with one layer ansatz,
the limited expressivity leads to results well below optimal ratio of $1.0$
even without noise. We employ the virtual node technique and readout error
mitigation. The Q-score ratios are averages over 100 random Maxcut problems.
We used 2048 shots per optimization step in QAOA.

##  5 Applications to research

Small-scale quantum computers have been used in many different areas in
scientific research including but not limited to physics, chemistry and
mathematics. Current trend in NISQ computer research is, no doubt, toward
scaling up physical qubits for commercial use of quantum computers.
Nonetheless, [45] reports that there are many research papers published in
scientific journals, which demonstrate “proofs of principle” of scientific
ideas with a small-scale quantum computer.

We illustrate three such examples from physics, mathematics and chemistry in
this section, and demonstrate them with our 5-qubit superconducting quantum
computer.

###  5.1 Simulating Neutrino Oscillations

It is known that there are at least three types (i.e. flavors) of neutrinos in
Nature. They are called $\nu_{e},\nu_{\mu}$ and $\nu_{\tau}$. Masses of these
neutrinos are not diagonal in the flavor basis
$\\{|\nu_{e}\rangle,|\nu_{\mu}\rangle,|\nu_{\tau}\rangle\\}$. Let us call the
eigenvectors of the mass matrix as
$\\{|\nu_{1}\rangle,|\nu_{2}\rangle,|\nu_{3}\rangle\\}$ with masses
(eigenvalues) $m_{1}<m_{2}<m_{3}$. Let $H$ be the Hamiltonian that describes
neutrinos. The mass eigenstates satisfy

$H|\nu_{k}\rangle=E_{k}|\nu_{k}\rangle\quad(k=1,2,3)$ (17)

where $E_{k}=\sqrt{p^{2}c^{2}+m_{k}^{2}c^{4}}$, $p$ is the momentum of the
neutrino, and $c$ is the velocity of light.

These two basis states are related by a unitary matrix called the
Pontecorvo–Maki–Nakagawa–Sakata matrix
$U_{\mathrm{PMNS}}=(\langle\nu_{\alpha}|\nu_{j}\rangle)$ [46, 47, 48] as

$\begin{pmatrix}\nu_{e}\\\ \nu_{\nu}\\\ \nu_{\tau}\\\
\nu_{X}\end{pmatrix}=U_{\mathrm{PMNS}}\begin{pmatrix}\nu_{1}\\\ \nu_{2}\\\
\nu_{3}\\\ \nu_{4}\end{pmatrix}$ (18)

where

$U_{\mathrm{PMNS}}=\begin{pmatrix}0.8255&0.5445&-0.142+0.0434i&0\\\
-0.2709+0.02739i&0.6057\,+0.0181i&0.7475&0\\\
0.4938\,+0.0237i&-0.5798+0.0157i&0.6475&0\\\ 0&0&0&1\end{pmatrix}.$ (19)

Here fictitious neutrinos $\nu_{X}$ and $\nu_{4}$ are introduced so that this
system can be simulated with a two-qubit system. $\nu_{X}$ and $\nu_{4}$ are
decoupled from the physical neutrinos and have no physical significance. This
fourth neutrino may be utilized in a theory with an exotic neutrino which is
yet to be discovered.

We closely follow [49] in the following. We keep the CP-violating phase
$\delta_{\mathrm{CP}}$ in $U_{\mathrm{PMNS}}$ while [49] ignored this phase.
The phase was taken into account in [50], which also generalizes the
simulation to arbitrarily many neutrino species. We employ parameters
announced in November 2022 [48] in Eq. (19).

Suppose $|\nu_{\mu}\rangle=(0,1,0,0)^{t}$ is created at $(x,t)=(0,0)$. The
probability of detecting $|\nu_{\alpha}\rangle$ $(\alpha=e,\mu,\tau)$ at $t>0$
is

$\displaystyle p_{\alpha}(t)$ $\displaystyle=$
$\displaystyle|\langle\nu_{\alpha}|\nu_{\mu}(t)\rangle|^{2}=|\langle\nu_{\alpha}|e^{-iHt/\hbar}|\nu_{\mu}\rangle|^{2}$
(20) $\displaystyle=$
$\displaystyle\left|\sum_{k=1}^{4}\langle\nu_{\alpha}|\nu_{k}\rangle
e^{-iE_{k}t/\hbar}\langle\nu_{k}|\nu_{\mu}\rangle\right|^{2}$ $\displaystyle=$
$\displaystyle|\langle\nu_{\alpha}|U_{\mathrm{PMNS}}\
\mathrm{diag}(e^{-iE_{1}t/\hbar},e^{-iE_{2}t/\hbar},e^{-iE_{3}t/\hbar},e^{-i\phi})\
U_{\mathrm{PMNS}}^{\dagger}|\nu_{\mu}\rangle|^{2},$

where $\phi$ is an unphysical phase.

Let us simulate this system with a quantum computer. There are three steps to
take.

  * •

Express $|\nu_{\mu}\rangle=|01\rangle$ in terms of the mass eigenstates
$|\nu_{k}\rangle$, which is done by applying $U_{\mathrm{PMNS}}^{\dagger}$ on
$|\nu_{\mu}\rangle$.

  * •

Apply the time-evolution operator $V(t)=e^{-iHt/\hbar}$ on this state to find
$|\nu_{\mu}(t)\rangle=V(t)U_{\mathrm{PMNS}}^{\dagger}|\nu_{\mu}\rangle$.
$V(t)$ is decomposed into two one-qubit gates as $V(t)=S_{1}(t)\otimes
S_{2}(t)$, whose explicit forms are given below.

  * •

Measure $|\nu_{\mu}(t)\rangle$ in the basis
$\\{|\nu_{e}\rangle,|\nu_{\mu}\rangle,|\nu_{\tau}\rangle\\}$. This is done by
applying $U_{\mathrm{PMNS}}$ on $|\nu_{\mu}(t)\rangle$ and measure the output
with binary basis.

Let us analyze these steps in depth. We prepare the initial state
$|\nu_{\mu}\rangle=|01\rangle$ in the $\\{|\nu_{k}\rangle\\}$ basis as

$U_{\mathrm{PMNS}}^{\dagger}|10\rangle=u_{\mu 1}^{*}|\nu_{1}\rangle+u_{\mu
2}^{*}|\nu_{2}\rangle+u_{\nu 3}^{*}|\nu_{3}\rangle,$ (21)

where $u_{\mu k}$ are the matrix elements of $U_{\mathrm{PMNS}}$.

The time-evolution operator $V(t)$ is diagonal in this basis and takes the
form

$V(t)=\mathrm{diag}(e^{-iE_{1}t/\hbar},e^{-iE_{2}t/\hbar},e^{-iE_{3}t/\hbar},e^{-i\phi}).$
(22)

Since the overall phase has no physical significance, we may factor out
$e^{-iE_{1}t/\hbar}$ so that

$V(t)=\mathrm{diag}(1,e^{-iE_{21}t/\hbar},e^{-iE_{31}t/\hbar},e^{-i\phi^{\prime}})=S_{1}(t)\otimes
S_{2}(t),$ (23)

where $E_{21}=E_{2}-E_{1}$, $E_{31}=E_{3}-E_{1}$ and $\phi^{\prime}$ is
another unphysical phase. Here

$S_{1}(t)=\begin{pmatrix}1&0\\\ 0&e^{-iE_{31}t/\hbar}\end{pmatrix},\quad
S_{2}(t)=\begin{pmatrix}1&0\\\ 0&e^{-iE_{21}t/\hbar}\end{pmatrix},$

where we took advantage of the fact that $\phi^{\prime}$ has no physical
meaning. Now we have $|\nu_{\mu}(t)\rangle$, the state of neutrino at $t$,
which was $|\mu_{\nu}\rangle$ at $t=0$, as

$|\nu_{\mu}(t)\rangle=(S_{1}(t)\otimes
S_{2}(t))U_{\mathrm{PMNS}}^{\dagger}|01\rangle.$ (24)

Since the neutrino masses are very small and the velocity is very close to the
speed of light $c$, we may approximate
$E_{k}=\sqrt{p^{2}c^{2}+m_{k}^{2}c^{4}}$ as $E_{k}\simeq
pc+m_{k}^{2}c^{3}/2p$. Then

$E_{k1}\simeq\frac{\Delta m_{k1}^{2}c^{3}}{2p}\quad(k=2,3)$ (25)

where $\Delta m_{k1}^{2}=m_{k}^{2}-m_{1}^{2}$. We employ $\Delta
m_{21}^{2}=7.39\times 10^{-5}\mathrm{eV}^{2}$ and $\Delta
m_{31}^{2}=2.45\times 10^{-3}\mathrm{eV}^{2}$ in our analysis [51].

By approximating $E\simeq pc$ and $L\simeq ct$, we have

$e^{-iE_{k1}t/\hbar}\simeq\exp\left(-i\frac{\Delta
m_{k1}^{2}c^{3}}{2\hbar}\frac{L}{E}\right).$ (26)

The exponent is expressed numerically with physical units as

$\frac{\Delta m_{k1}^{2}c^{3}}{2\hbar}\frac{L}{E}\simeq 2.534\,\Delta
m_{k1}^{2}\,[\mathrm{eV}^{2}]\times\frac{L\,[\mathrm{km}]}{E\,[\mathrm{GeV}]}.$

These steps are implemented with a quantum computer with different $t$, namely
different $L/E$ and the results are compared with theoretical prediction.
Figure 15 (a) shows the quantum circuit for this scheme while (b) shows the
gate decomposition of $U_{\mathrm{PMNS}}$. The overall gate decomposition
depends on $L/E$. Figure 15 (c) shows the transpiled gate decomposition for
$L/E=4000$ [km/GeV].

(a)

(b)

(c)

Figure 15: (a) Quantum circuit to simulate neutrino oscillation among 3
generations. The input state is $|01\rangle=|\nu_{\mu}\rangle$. (b)
Implementation of $U_{\mathrm{PMNS}}$ using our native gates. (c) Circuit to
simulate neutrino oscillations, transpiled to our native gates at $L/E=4000$.
The number of gates is reduced via circuit optimization during the
transpilation. Note that in (b) and (c) the order of qubits is reversed
according to the Qiskit convention. The second parameter of $R$ is defined
$\text{mod}~{}2\pi.$

Figure 16 shows the theoretical curves predicting the detection probabilities
of three neutrinos and our quantum computer output as functions of $L/E$.

Figure 16: Predicted measurement probabilities of neutrino species (solid
curves) and measured probabilities (markers) as functions of
$L/E~{}[\mathrm{km/GeV}]$, where $\nu_{\mu}$ is created at $L=0$ at $t=0$. We
used 5000 shots per data point and applied readout error mitigation.

Probabilities oscillate as a function of $L/E$ while the sum of probabilities
is always 1. This oscillation is possible only when different types of
neutrinos have different masses ($\Delta m_{k1}\neq 0$). Thus neutrino
oscillation is a smoking gun of massive neutrinos.222The “standard model” of
particle physics had no neutrino mass terms for many decades. Neutrino
oscillation was the subject of the Nobel Prize in Physics in 2015 [52].

###  5.2 Estimation of the Jones polynomials

Knots, links and braids are fascinating subjects of topology. An unexpected
encounter between mathematics (topology) and physics (statistical physics) was
discovered by Vaughan Jones in 1984. He discovered knot and link invariants,
later called the Jones polynomials, that characterize oriented knots and
links. His work was inspired by statistical mechanics of generalized spin
models defined on a lattice. Jones was awarded the Fields medal in 1990 for
his achievements including the discovery of the Jones polynomials.

Estimation of the Jones polynomials by employing a quantum computer was
proposed in [53, 54] and demonstrated with an NMR quantum computer [55]. We
reproduce here the results of [55] using our superconducting quantum computer.

We first introduce a braid $b$ associated with a link $L$. A link is an
embedding of a set of loops in $\mathbb{R}^{3}$ or the 3-dimensional sphere
$S^{3}$. If a link is made of one component, it is called a knot $K$. Let
$(x,y,t)$ be a coordinate of 3-dimensional space-time. A braid $b$ is a set of
strings that connect $n$ points $(0,0,0),(1,0,0),\ldots(n,0,0)$ on
$\mathbb{R}^{2}$ at $t=0$ and $(0,0,1),(1,0,1),\ldots(n,0,1)$ at $t=1$ without
intersecting or going backwards in time. Alexander’s theorem claims that every
knot and link can be expressed as a braid whose end points are closed, see
Fig. 17.

Figure 17: (a) Hopf link and its corresponding braid. (b) Trefoil knot and its
corresponding braid. Braids are closed with dotted lines to form the link and
the knot.

The set of braids with $n$ strands has a group structure called the Artin
group $B_{n}$, whose generators are denoted as $\sigma_{k}$. The generator
$\sigma_{k}$ twists the $k$th strand and $(k+1)$st strand as shown in Fig. 18
for $n=3$. The inverse $\sigma_{k}^{-1}$ twists them in the opposite
direction. The set of generators satisfy the following relations:

$\begin{array}[]{cl}\sigma_{k}\sigma_{k}^{-1}=1,&k=1,2,\ldots,n-1\\\
\sigma_{k}\sigma_{k+1}\sigma_{k}=\sigma_{k+1}\sigma_{k}\sigma_{k+1},&k=1,2,\ldots,n-2\\\
\sigma_{j}\sigma_{k}=\sigma_{k}\sigma_{j},&|j-k|\geq 2.\end{array}$ (27)

An arbitrary braid $b$ can be expressed in terms of successive applications of
these generators and their inverses as

$\sigma_{j_{p}}^{s_{p}}\sigma_{j_{p-1}}^{s_{p-1}}\ldots\sigma_{j_{2}}^{s_{2}}\sigma_{j_{1}}^{s_{1}},$
(28)

where $s_{k}\in\\{1,-1\\}$, $j_{k}\in\\{1,2,\ldots,n-1\\}$ and
$\sigma_{j_{1}}^{s_{1}}$ is applied first and $\sigma_{j_{p}}^{s_{p}}$ last.
This is called the braid word of $b$. Braid words are not unique and there are
infinitely many braid words corresponding to the same braid.

In the following, we are concerned with three-strand braids, namely $n=3$. It
has two generators $\sigma_{1}$ and $\sigma_{2}$ as shown in Fig. 18.

Figure 18: Generators $\sigma_{1},\sigma_{2}$ of three-strand braids.

One possible braid word of the trefoil knot (Fig. 17 (b)) is $\sigma_{1}^{3}$.
Closure of the three-strand braid $\sigma_{1}^{3}$ results in the trefoil and
the trivial knot, as shown in Fig. 19 (a).

Figure 19: (a) Simplest representation $\sigma_{1}^{3}$ of the trefoil knot.
(b) Equivalent but more knotted representation of the trefoil knot.

It is possible to “unwind” crossing of a braid diagram by introducing the
generators $\\{U_{i}\\}$ of the Temperley-Lieb algebra $\text{TL}_{n}$. There
are two generators $U_{1},U_{2}$ for $n=3$ and they satisfy

$U_{i}^{2}=\delta U_{i},\ U_{1}U_{2}U_{1}=U_{1},\ U_{2}U_{1}U_{2}=U_{2}$ (29)

where $\delta=-A^{2}-A^{-2}$ with $A=e^{i\theta}$ a complex number with unit
modulus. The map $\rho:B_{3}\to\mathrm{GL}_{2}(\mathbb{C})$ defined as

$\rho(\sigma_{i})=AI+A^{-1}U_{i}$ (30)

is a representation of $B_{3}$. We take the simplest braid word of the trefoil
$\sigma_{1}^{3}$ and its representation $\rho(\sigma_{1}^{3})$ here. Note that
the relations of the braid generators (27) are satisfied if $U_{i}$ satisfies
Eq. (29). The representation $\rho$ is unitary whenever $U_{i}$ is a real
symmetric matrix satisfying $\delta^{2}\geq 1$. The last condition is
satisfied if

$\theta\in[0,\pi/6]\sqcup[\pi/3,2\pi/3]\sqcup[5\pi/6,7\pi/6]\sqcup[4\pi/3,5\pi/3]\sqcup[11\pi/6,2\pi].$

Explicitly, the generators $U_{i}$ are given as

$U_{1}=\begin{pmatrix}\delta&0\\\ 0&0\end{pmatrix}\quad\mbox{and}\quad
U_{2}=\begin{pmatrix}\delta^{-1}&\sqrt{1-\delta^{-2}}\\\
\sqrt{1-\delta^{-2}}&\delta-\delta^{-1}\end{pmatrix}.$ (31)

We need to define $w(b)\in\mathbb{Z}$ called the writhe of a braid $b$ before
we introduce the Kauffman bracket and the Jones polynomial. Suppose a braid
word of $b$ is given as Eq. (28). Then the writhe of $b$ is defined as the sum
of the exponents,

$w(b)=\sum_{i=1}^{p}s_{i}.$ (32)

For the Hopf link and the trefoil knot we find $w(\mbox{Hopf link})=2$ and
$w(\mbox{trefoil})=3$, respectively.

The Kauffman bracket of the trefoil is obtained as

$\langle\mbox{trefoil}\rangle=\frac{\langle\bar{b}\rangle}{\delta}=\frac{1}{\delta}(\mathrm{tr}\rho(\sigma_{1}^{3})+A^{w(b)}(\delta^{2}-2))=-A^{5}-A^{-3}+A^{-7},$
(33)

where $b$ is the braid word given in Fig. 17 (b) and $\bar{\ }$ stands for the
closure of $b$. Note that $\bar{b}$ is made of the trefoil knot and the
trivial knot. The factor $1/\delta$ in Eq. (33) removes the contribution of
the trivial knot.

The Kauffman bracket of the Hopf link is obtained in a similar way as

$\langle\mbox{Hopf
link}\rangle=\frac{1}{\delta}(\mathrm{tr}\rho(\sigma_{1}^{2})+A^{2}(\delta^{2}-2))=-A^{4}-A^{-4}.$
(34)

The Kauffman bracket is invariant under the Reidemeister moves II and III but
not under the Reidemeister move I, and hence cannot be a knot invariant. The
Jones polynomial is obtained by multiplying the Kauffman bracket with
$(-A^{3})^{-w(b)}$ to make it invariant under all three Reidemeister moves.
For the trefoil knot, we obtain the Jones polynomial

$V_{\mathrm{trefoil}}(A)=(-A^{3})^{-3}(-A^{5}-A^{-3}+A^{-7})=A^{-4}+A^{-12}-A^{-16}.$
(35)

It is common to introduce $t=A^{-4}$ so that

$V_{\mathrm{trefoil}}(t)=-t^{4}+t^{3}+t.$ (36)

The Jones polynomial of the Hopf link is

$V_{\mathrm{Hopf\ link}}=-A^{-10}-A^{-2}=-\sqrt{t}(1+t^{2}).$ (37)

The Jones polynomial is a Laurent polynomial in $\sqrt{t}$ in general.

(c)

Figure 20: (a) Quantum circuit to estimate the Jones polynomial with input
$|0\rangle\langle 0|\otimes I_{2}/2$. (b) Quantum circuit to estimate the
Jones polynomial using ancilla qubit to synthesize the mixed state. The input
state is $|000\rangle$. To estimate the Jones polynomial, the expectation
values of $X_{1}$ and $Y_{1}$ are estimated for this state. (c) Circuit for
the trefoil knot, appended with a $H$ gate on the first qubit for measuring
$E(X_{1})$ and transpiled to our native gates and connectivity at
$\theta=\pi/6$. The second parameter of $R$ is defined $\text{mod}~{}2\pi.$

Let us use a quantum computer to estimate the trace
$\mathrm{tr}\rho(\sigma_{1}^{k})$ in the Kauffman bracket. Consider the
quantum circuit Fig. 20 (a) with $U=\rho(\sigma_{1}^{k})$ and the input state

$\rho_{0}=|0\rangle\langle
0|\otimes\frac{1}{2}I_{2}=\frac{1}{2}\begin{pmatrix}I_{2}&0\\\
0&0\end{pmatrix},$

where $I_{2}/2$ is the maximally mixed state. The state after quantum circuit
is applied is

$\rho_{1}=\begin{pmatrix}I_{2}&0\\\
0&U\end{pmatrix}\frac{1}{4}\begin{pmatrix}I_{2}&I_{2}\\\
I_{2}&I_{2}\end{pmatrix}\begin{pmatrix}I_{2}&0\\\
0&U^{\dagger}\end{pmatrix}=\frac{1}{4}\begin{pmatrix}I_{2}&U^{\dagger}\\\
U&I_{2}\end{pmatrix}.$

The expectation value of $X_{1}$ with respect to $\rho_{1}$ is

$E(X_{1})=\mathrm{tr}(X_{1}\rho_{1})=\frac{1}{2}\mathrm{Re\ tr}\,U$ (38)

while the expectation value of $Y_{1}$ is

$E(Y_{1})=\mathrm{tr}(Y_{1}\rho_{1})=\frac{1}{2}\mathrm{Im\ tr}\,U.$ (39)

Hence, the $\mathrm{tr}\,U$ is found from estimating these two expectation
values.

The above scheme fits well with NMR quantum computer, in which the system is
in a maximally mixed state with a good approximation. A superconducting
quantum computer is ideally in a pure state and the above scheme cannot be
applicable in its original form. We use purification to “synthesize” a
uniformly mixed state from a pure state for this purpose. Let us consider the
circuit in Fig. 20 (b). The bottom two qubits are in the Bell state

$|\Phi_{+}\rangle=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)$

after application of the Hadamard gate and the CNOT gate. The middle qubit is
in a maximally mixed state if the bottom qubit is ignored (i.e. partially
traced out). We have already seen this in section 4.2.2. Observe that

$\mathrm{tr}_{2}|\Phi_{+}\rangle\langle\Phi_{+}|=\frac{1}{2}I_{2}.$

Then the output of the top two qubits is the same as that of Fig. 20 (a).

Suppose the quantum circuit Fig. 20 (b) is applied on $|000\rangle$. Then the
output state is

$|\Psi\rangle=\frac{1}{2}(|000\rangle+|011\rangle+|1\rangle(U|0\rangle)|0\rangle+|1\rangle(U|1\rangle)|1\rangle).$
(40)

We estimate the expectation value of $X_{1}$ with respect to $|\Phi\rangle$ to
get

$E(X_{1})=\langle\Phi|X_{1}|\Phi\rangle=\frac{1}{4}(\mathrm{tr}\,U+\mathrm{tr}\,U^{\dagger})=\frac{1}{2}\mathrm{Re}(\mathrm{tr}\,U),$
(41)

similarly for $Y_{1}$ we get

$E(Y_{1})=\langle\Phi|Y_{1}|\Phi\rangle=\frac{i}{4}(-\mathrm{tr}\,U+\mathrm{tr}\,U^{\dagger})=\frac{1}{2}\mathrm{Im}(\mathrm{tr}\,U),$
(42)

from which we estimate $\mathrm{tr}\,U$. Figure 20 (c) shows the transpiled
circuit to estimate $E(X_{1})$ for $\theta=\pi/6$.

Figure 21 shows the real and the imaginary parts of
$\mathrm{tr}\,\rho(\sigma_{1}^{k})$ obtained using $|\Phi\rangle$ with $k=2$
for the Hopf link while $k=3$ for the trefoil knot. The readout error
mitigated results are marked by the blue circles.

When quantum circuits get deeper, consisting of several layers of gates,
errors that happen in the bulk of the circuits start to accumulate and can
significantly affect the results. This is the case for the transpiled circuits
that we are considering here, see Fig. 20 (c), and we thus decided to
implement also error suppression and additional error mitigation techniques,
namely randomized compiling and zero-noise extrapolation. The mitigated
expectation values are shown with green diamonds and, in general, are indeed
closer to the ideal noiseless results (red). In more detail, we generate 30
different randomized compilations of the original circuit and measure each
20,000 times. The higher shot count is employed to combat the increased
variance resulting from zero-noise extrapolation, where we scale the noise by
factors of 3 and 5 using global folding. Polynomial fitting is used to
extrapolate to the limit of zero noise.

(a)

(b)

(c)

(d)

Figure 21: Trace in the Jones polynomial of the Hopf link and the trefoil
knot. (a) Real part and (b) imaginary part of the trace in the Hopf link,
estimated as the expectation values of $X$ and $Y$ on the first qubit,
respectively. Similarly, (c) and (d) show the real and the imaginary parts of
the trace for the trefoil knot. Solid curves are theoretical results while the
markers show experimental results evaluated on our quantum computer. Circles
are from readout error mitigated execution and diamonds include also
randomized compiling (RC) and zero-noise extrapolation (ZNE).

The Kauffman and the Jones polynomials may be obtained in many different ways.
All of them are easy if the trefoil is represented as in Fig. 19 (a), for
example. But it will be more demanding if the representation is more knotted
as in Fig. 19 (b). A typical classical evaluation of these polynomials
involves a sum over “states” obtained by splitting each crossing in two
different ways. There are $2^{m}$ states if there are $m$ crossings and the
task is exponentially hard as $m$ increases. In contrast, in quantum
computing, a controlled unitary gate is assigned to each crossing, i.e., the
braid group generator, which requires merely $m$ controlled unitary gates.

###  5.3 Embedding Techniques for Quantum Chemistry

Embedding techniques are theoretical frameworks used in quantum chemistry and
condensed matter physics to study the electronic structure of strongly
correlated materials. These methods are crucial and are particularly valuable
for systems where the effects of electron-electron interactions are
significant and where traditional methods such as density functional theory
(DFT) [56, 57] fail to provide accurate descriptions. They represent a
powerful tool to study the electronic properties of materials such as
transition-metal-oxides and rare-earth compounds. Embedding techniques have
been successfully applied to further the understanding of complex phenomena
such as metal-insulator transitions, magnetically ordered states and high-
temperature superconductivity.

The central idea of these techniques is to map a complex quantum many-body
problem to a self-consistent Anderson impurity model (AIM) [58], which
consists of a strongly correlated subsystem (impurity) and a weakly correlated
or non-correlated subsystem (bath). The impurity is treated more accurately
using a method capable of handling strong correlations, while the bath is
treated at a lower level of theory, often using mean-field approximations.
These descriptions of the impurity and the bath are combined in a self-
consistent loop over either single particle (i.e. density matrix embedding
theory (DMET) [59, 60] and rotationally invariant slave-boson (RISB)
techniques [61, 62, 63]) or two particle quantities (i.e. self-energy
embedding theory (SEET) [64, 65], dynamical mean-field theory (DMFT) [66, 67,
68] and its cluster extensions [67, 69]) in order to provide a more accurate
description of the entire system. Figure 22 is a schematic of the self-
consistent loop in embedding based techniques.

Figure 22: Schematic diagram of the embedding loop showcasing the part of the
loop that is to be performed on the quantum computer.

Consider the electronic structure (ES) Hamiltonian in second-quantized form:

$\mathcal{H}_{\text{ES}}=\sum_{pq}h^{pq}a^{\dagger}_{p}a_{q}^{\phantom{\dagger}}+\sum_{pqrs}h^{pqrs}a^{\dagger}_{p}a^{\dagger}_{q}a_{r}^{\phantom{\dagger}}a_{s}^{\phantom{\dagger}}$
(43)

where $p,q,r,s$ are indices of a given basis set and include the respective
spin indices $p\equiv p(\sigma)$. Further, $a^{\dagger}$ and $a$ are the
fermionic creation and annihilation operators and $h^{pq}$, $h^{pqrs}$ are
constants called the one- and two-electron integrals, respectively. Such a
Hamiltonian is then mapped to an Anderson impurity-bath model given by:

$\displaystyle\mathcal{H}_{\text{AIM}}=\sum_{k,\sigma}\varepsilon_{k}c_{k\sigma}^{\dagger}c_{k\sigma}^{\phantom{\dagger}}+\sum_{\sigma}(\varepsilon_{d}-\mu)d_{\sigma}^{\dagger}d_{\sigma}^{\phantom{\dagger}}+Un_{d\uparrow}^{\phantom{\dagger}}n_{d\downarrow}^{\phantom{\dagger}}+\sum_{k,\sigma}(V_{k}c_{k\sigma}^{\dagger}d_{\sigma}^{\phantom{\dagger}}+\text{H.c.}).$
(44)

where $k$ represents the summation index for the bath operators, $\sigma$
labels the spin, $d$ corresponds to the impurity operator with
$n_{d\sigma}=d^{\dagger}_{\sigma}d_{\sigma}$ and $c$ corresponds to the
operators on the non-interacting bath, $\varepsilon_{d/c}$ correspond to the
onsite-energy of the impurity and bath, respectively, with the chemical
potential $\mu$. In spite of giving a simple description of the lattice
problem, this model is by itself challenging to solve for state-of-the-art
numerical techniques such as density matrix renormalization group (DMRG) [70,
71, 72, 73], quantum Monte Carlo (QMC) [74, 75, 76, 77] and others. It has
been recently shown that a promising alternative approach for computing the
ground states of such systems comes from using variational algorithms
implemented on quantum devices [78, 79] one of which, QAOA, has already been
discussed in section 4.3.

The variational quantum eigensolver (VQE) [80, 81, 82] is another quantum
algorithm designed for finding the ground state energy, which is a fundamental
quantity for quantum systems of interest in condensed matter physics and
quantum chemistry. The basic idea behind VQE is to use a hybrid approach where
the optimization of variational parameters performed on a classical computer
is combined with measurements on the quantum computer to find an approximation
to the ground state energy. The algorithm involves mainly a trial wavefunction
in the form of a parameterized quantum circuit, chosen to represent a guess
for the ground state of the quantum system. This circuit, controlled by a set
of variational parameters, is executed on a quantum computer. Measurements are
made on the final prepared state to calculate the expectation value of the
Hamiltonian of the system. Then classical optimization algorithms are employed
to tune the variational parameters in order to minimize the expectation value
of the Hamiltonian. This process is repeated iteratively until the ground
state energy is sufficiently minimized and the solution converges.

(a)

(b)

Figure 23: (a) Hardware efficient Ansatz used for the VQE. (b) Comparison of
the VQE energy as a function of the iterations for convergence of an Anderson
impurity model at $U=2$ on our 5-qubit quantum computer with and without
readout error mitigation. The exact energy is also shown for comparison.

On our 5-qubit quantum computer (see Fig. 23) we aim to perform a VQE
calculation in order to approximate the ground state of the one-impurity-site
and one-bath-site AIM ($k=1$ in Eq. (44)). The fermionic Hamiltonian of the
AIM has to be mapped to a qubit Hamiltonian using the following Jordan-Wigner
transformation [83]:

$\displaystyle d_{\uparrow}^{\dagger}$
$\displaystyle=\sigma_{0}^{-}=\frac{1}{2}(X_{0}-iY_{0})$ (45) $\displaystyle
c_{1\uparrow}^{\dagger}$
$\displaystyle=Z_{0}\sigma_{1}^{-}=\frac{1}{2}Z_{0}(X_{1}-iY_{1})$ (46)
$\displaystyle d_{\downarrow}^{\dagger}$
$\displaystyle=Z_{0}Z_{1}\sigma_{2}^{-}=\frac{1}{2}Z_{0}Z_{1}(X_{2}-iY_{2})$
(47) $\displaystyle c_{1\downarrow}^{\dagger}$
$\displaystyle=Z_{0}Z_{1}Z_{2}\sigma_{3}^{-}=\frac{1}{2}Z_{0}Z_{1}Z_{2}(X_{3}-iY_{3})$
(48)

which leads to the qubit Hamiltonian

$\displaystyle\mathcal{H}_{\text{AIM}}^{\text{JW}}$
$\displaystyle=(\varepsilon_{d}+\varepsilon_{1}-2\mu)-\frac{1}{2}(\varepsilon_{d}-\mu+2U)(Z_{0}+Z_{2})-\frac{1}{2}\varepsilon_{1}(Z_{1}+Z_{3})$
$\displaystyle+\frac{1}{4}UZ_{0}Z_{2}+\frac{1}{2}V_{1}(X_{0}X_{1}+Y_{0}Y_{1}+X_{2}X_{3}+Y_{2}Y_{3}).$
(49)

Grouping Hamiltonian terms that have support on different qubits allows us to
reduce the number of measurements performed. For example an
$X_{0}X_{1}X_{2}X_{3}$ measurement on the Ansatz state can be used to compute
both $X_{0}X_{1}$ and $X_{2}X_{3}$ terms of the Hamiltonian. A hardware
efficient Ansatz as shown in Fig. 23 (a) is chosen. Since the problem requires
only four qubits, the best qubits based on fidelity are chosen from the
hardware to perform the calculations and keeping $0^{th}$ qubit of the spin
Hamiltonian fixed at the center of the star topology. Parameterized single
qubit $R_{y}$ gates are introduced and a total of seven parameters are tuned
by the L-BGFS-B classical optimizer from Scipy-optimizer package [84]. This is
a gradient based optimizer where the derivative is computed using the
parameter shift rule [85, 86]. One has:

$\displaystyle\frac{\partial
E_{{\bm{\theta}}}}{\partial\theta_{i}}=\frac{E_{\theta_{i}+\frac{\pi}{2}}-E_{\theta_{i}-\frac{\pi}{2}}}{2}$
(50)

where
$E_{{\bm{\theta}}}=\langle\psi({\bm{\theta}})|\mathcal{H}_{\text{AIM}}^{\text{JW}}|\psi(\bm{\theta})\rangle$
with ${\bm{\theta}}=\\{\theta_{1},\theta_{2}\cdots\\}$. Figure 23 (b) shows
the energy measurements for different iterations computed without and with
read-out error mitigation at $\epsilon_{d}=\mu,\epsilon_{1}=0,V=1$ and $U=2$.
We have used 5000 shots for each measurement. The results can be further
improved using advanced error mitigation strategies, such as zero-noise
extrapolation (ZNE) [87, 88, 89] or probabilistic error cancellation (PEC)
[90, 91, 92]. The converged parameterized state can be used to compute
quantities such as density matrix or the Green’s function which are necessary
components of some of the aforementioned embedding techniques.

VQE has been proposed as a promising algorithm for near-term quantum computers
to solve certain models from quantum chemistry and condensed matter physics.
It is important to note though, that practical implementations of VQE are
still limited by the capabilities of available quantum hardware, and there is
active ongoing work on improving and refining these algorithms. Nevertheless,
it serves as a gateway to explore and investigate advanced embedding
techniques on the currently available quantum hardware.

##  6 Summary

An on-site quantum computer can be utilized in education and research for
quantum computing, quantum information and quantum theory. We have
demonstrated some of these topics with the 5-qubit superconducting IQM SparkTM
prototype in this paper.

First, we presented the tools and programs used. Then, we showed how they can
be used in education and research. Certain demonstrations, such as calibration
and working with qutrits, are only possible with an on-site quantum computer.
It is also used to explore the complex quantum realm and reproduce recent
breakthroughs in mathematics, physics, and chemistry for a better
understanding.

##  7 Discussion

This physical on-site quantum computer is vital for making quantum computing
accessible to more people and talents, teaching quantum concepts, and
enhancing our understanding of quantum theory and computing. It can also be
used for research, as we have shown in Section 5. A recent survey reveals that
many research papers demonstrating “proofs of principle” used a quantum
computer with five qubits of fewer [45].

We anticipate that in the very near-term future, every leading university and
research institute, wanting to stay competitive in quantum computing education
and research, will have physical access to affordable on-site quantum
computers, such as the IQM SparkTM [6].

## Acknowledgments

The work of the IQM technology team is acknowledged for having created the
technology used here in the educational and research setting. We would like to
thank Roberto Moretti and Michihisa Wakui for enlightening discussions. We are
grateful to Andrew Guthrie for careful reading of the manuscript.

## Funding

The whole work is funded by IQM Quantum Computers.

## Availability of data and materials

The codes employed in this paper are available from the corresponding author
upon request.

## Competing interests

IQM SparkTM is a product of IQM, the employer of all authors

## Authors’ contributions

All authors contributed equally in this paper.

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