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# An Improved Volumetric Metric for Quantum Computers via more Representative
Quantum Circuit Shapes.

Keith Miller  Systems Engineering Innovation Center, MITRE, 7515 Colshire
Drive, McLean, Virginia 22102, USA  Homeland Security Systems Engineering and
Development Institute (HSSEDI)™ operated by the MITRE Corporation
<EMAIL_ADDRESS>Charles Broomfield<EMAIL_ADDRESS>Homeland Security
Systems Engineering and Development Institute (HSSEDI)™ operated by the MITRE
Corporation  Ann Cox  Department of Homeland Security, Science and Technology
Directorate, USA<EMAIL_ADDRESS>Joe Kinast  Quantum Technologies Group,
MITRE, 200 Forrestal Rd. Princeton, New Jersey 08540, USA  Homeland Security
Systems Engineering and Development Institute (HSSEDI)™ operated by the MITRE
Corporation<EMAIL_ADDRESS>Brandon Rodenburg  Quantum Technologies Group,
MITRE, 200 Forrestal Rd. Princeton, New Jersey 08540, USA  Homeland Security
Systems Engineering and Development Institute (HSSEDI)™ operated by the MITRE
Corporation<EMAIL_ADDRESS>
###### Abstract

In this work, we propose a generalization of the current most widely used
quantum computing hardware metric known as the quantum volume [1, 2]. The
quantum volume specifies a family of random test circuits defined such that
the logical circuit depth is equal to the total number of qubits used in the
computation. However, such square circuit shapes do not directly relate to
many specific applications for which one may wish to use a quantum computer.
Based on surveying available resource estimates for known quantum algorithms,
we generalize the quantum volume to a handful of representative circuit
shapes, which we call Quantum Volumetric Classes, based on the scaling
behavior of the logical circuit depth (time) with the problem size (qubit
number).

As a technology, quantum computing is in its infancy but developing rapidly.
In the near term, noisy and intermediate-scale quantum (NISQ) systems may
become useful for specific niche applications [3]. In the long term, with the
development of fault-tolerant (FT) systems, this technology is expected be
extremely disruptive and transformative. Clear metrics to evaluate this
technology are crucial for evaluating and comparing performance of various
quantum devices and platforms in the near term, as well as creating
quantitative tools to better anticipate more long-term disruptions of this
technology.

An ideal metric has several key features, which we list in Table 1 below.
First, we want such a metric to be defined in such a way that it is universal
across potential quantum computers. This means we want a metric that is
defined at the logical computational level, independent of the underlying
physical platform. In addition, we want a metric that is universal across
maturity levels. We do not just want a metric for NISQ devices, but one that
applies equally to both NISQ and more mature FT systems that utilize error
correction.

  1. 1.

Universal and platform independent

  2. 2.

Applicable to both near (NISQ) and long term (FT) systems

  3. 3.

Simple enough to be useful and understandable to non-experts

  4. 4.

Representative of the computational power needed to execute quantum algorithms

Table 1: Features of an ideal metric

We also require the metric to be simple and understandable enough to be
useful, including to non-experts. There will inevitably be trade-offs between
precision and simplicity. However, one of the key reasons for high-level
metrics is to provide some level of guidance to non-experts, whether they are
end users, application specialists, leadership within a company, or a
government organization. Such potential stakeholders need quantitative tools
to make strategic decisions or to track the technology over time.

Finally, the ideal metric needs to represent the computational power in a
straightforward way, i.e., a bigger number represents a more powerful device.
These values should also be tied closely to applications, making it possible
to more quickly determine if a device can run some specific application.
Conversely, having a metric closely aligned with the application space allows
specific applications to be framed directly in terms of this framework. This
makes any such system far more useful to potential end users.

The rest of this paper is laid out as follows. Section 1 gives a detailed
description of the quantum volume metric, including its shortcomings. In
particular, there are two fundamental issues. First, the quantum volume does
not obviously relate to many of the applications that one may wish to use a
quantum computer for in terms of resources needed to perform a computation.
The second is the fact that the quantum volume does not explicitly take into
account quantum error correction, a feature that will be necessary for this
technology to mature beyond the NISQ era that exists today. This is especially
important in light of the rapid progress towards implementing error correction
below the fault-tolerant threshold [4, 5, 6, 7, 8, 9, 10, 11].

##  1 Quantum Volume

One of the most widespread high-level metrics currently in use is the quantum
volume, originally introduced by IBM in 2017 [1, 2]. The quantum volume is
meant to capture the ability of a quantum computing device to prepare any
state from a random sampling of the state space of a set of qubits. A quantum
computer with $n$ qubits is represented by a $2^{n}$-dimensional Hilbert
space. However, if the qubits are poorly controlled or subject to excess
noise, the device cannot effectively sample the full state space of the $n$
qubits.

The ability to randomly access any part of the $2^{n}$-dimensional state space
of $n$ qubits, or equivalently the ability to randomly scramble an initial
state, requires a gate depth of $\mathcal{O}(n)$ [12]. For this reason, the
quantum volume of a device was defined as the largest square circuit that the
device can implement. Mathematically this can be writted as [2]

$\log_{2}(V_{Q})=\max_{n<n_{\text{max}}}\left(\min\left[n,d\right]\right),$
(1)

where $n$ is the subset of the $n_{\text{max}}$ qubits available used in the
circuit and $d$ is the circuit depth111The quantum volume was initially
defined as $V_{Q}=\max_{n}\left(\min\left[n,d\right]^{2}\right),$ representing
the space-time volume of a square $n\times n$ circuit [1]. This was later
changed to the exponential definition in [2]. However, due to the exponential
scaling of $V_{Q}$ with $n$, this number is usually quoted as
$\log_{2}(V_{Q})$ instead.. Each of the $d$ layers of the algorithm consists
of a random permutation of the qubits followed by the pairwise application of
a random 2-qubit SU(4) matrix (see Figure 1). The qubit permutation may
consist of sets of swap gates to move qubit states around, or a simple logical
relabeling of the qubits without applying any additional gates to the extent
the qubit layout and connectivity allows.

Figure 1: A circuit diagram for benchmarking the quantum volume. The circuit
consists of $d$ layers of a random permutation of the qubits (represented by
$\pi$) followed by random two-qubit SU(4) gates.

In order to determine the quantum volume for a specific hardware platform, a
definition of success is needed for a given test circuit. The original quantum
volume benchmark uses the heavy output criteria where so called “heavy
outputs” are measured with a probabily $>2/3$ [2, 13]. A downside of this
criteria is that probability outputs must be computed in advance, a problem
that is generally computationally hard classically. However, it is possible to
construct benchmark circuits that scale efficiently enough to be useful for
large-scale quantum circuits [14].

A problem with the quantum volume metric is the reliance on defining quantum
volume as an exponential, e.g. $2^{n}$, rather than just quoting the integer
$n$ itself. This means that large-scale powerful systems will have
exponentially large values due to the fact that incremental improvements lead
to large differences in the quantum volume. Therefore, the quantum volume
fails to be a fair representation of computational power (item four in Table
1). For this reason, the logarithm of the quantum volume is generally quoted
rather than the volume itself. In this spirit, we also adopt this convention
when considering a generalized metric.

A second problem with the quantum volume is that as defined the quantum volume
does not explicitly specify how quantum error correction should be handled
when determining the metric value of a specific device, i.e., whether a
quantum volume test circuit is defined at the physical or logical level. As
such, the quantum volume is only applicable to the NISQ devices that currently
exist which do not utilize error correction. Therefore, this metric fails the
universality condition (item two) of Table 1. Given our emphasis on trying to
relate computational power to known algorithms (e.g., by looking for
representative circuit shapes based on known algorithms), we choose to define
our metric at the logical level. By this we mean that circuit shapes that
define our metric are defined in terms of logical resources, and the actual
physical implementation of this circuit incorporates as much (or as little)
quantum correction overhead as needed to successfully implement the circuit in
question. Ideally, one would also like to have an idea of how the
computational resources defined at the logical level scale with the physical
features (e.g., qubit number and physical error rates), however, this type of
resource estimation is beyond the scope of this current work.

A final problem that we see with the quantum volume is that a square $n\times
n$ circuit represents a minimum necessary circuit depth needed for many
potential quantum algorithms of interest (see section 2). This means that such
a circuit represents at best the first step in a full computation. In fact,
many quantum algorithms have significantly greater circuit depth than qubit
number. That makes the quantum volume a poor representative as a stand-in for
the capability of the device (item four in Table 1).

A generalization of the metric that instead considers the qubit number and
circuit depth as separate independent resources has been proposed [15].
Although more robust, this generalization comes at the cost of much greater
complexity, as a quantum computer is no longer represented by a single
quantitative value, but rather an entire family of successful circuit shapes
and sizes. Therefore, this full generalization although more descriptive and
thus more useful as a generalized framework for benchmarking [16], fails our
simplicity criteria as an ideal metric by itself (item three in Table 1).

In order to retain the advantages of this generalized framework [15] we look
at a restricted subset of possible circuit shapes. In this way we can balance
the simplicity of a few numbers representing our metric, while not restricting
ourselves to the orginal square circuit shape. To this end, a survey of
resource requirements for known quantum algorithms was performed in order to
determine typical circuit shapes to use in defining a generalized metric. The
results of this survey and the proposed circuit shapes are discussed in
section 2.

##  2 Identifying Volumetric Shapes

Quantum volume provides meaningful insight about a quantum computer’s ability
to implement algorithms which require a gate depth that scales no faster than
linearly in relation to number of qubits. In reality, however, several
algorithms appear to require a gate depth that scales faster than linearly in
relation to number of qubits. Thus, quantum volume has limited utility as a
universal metric for comparing the performance of quantum computers.

Application Area | Quantum Computing Era
---|---
NISQ |  FT |  All
Machine Learning | 11 | 13 | 24
Optimization | 9 | 5 | 14
Many-Body Physics/Chemistry | 9 | 7 | 16
Quantum Data Hiding | 0 | 6 | 6
Numerical Solvers | 0 | 3 | 3
Other | 0 | 2 | 2
Table 2: Breakdown of the algorithms with known resource estimates considered
in this paper [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32,
33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51,
52, 53, 54, 55, 56, 57, 58, 59, 60, 61]. We note that the total count of all
algorithms exceeds 58 due to the fact that some algorithms apply to both
machine learning and optimization.

In order to look for more useful quantum circuit shapes, we performed a survey
of known quantum algorithms with available resource estimates. Data collection
included algorithms designed for both NISQ and FT devices. After reviewing
over 225 research papers, we found 58 [17, 18, 19, 20, 21, 22, 23, 24, 25, 26,
27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45,
46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61] algorithms
with resource estimates that could be used to approximate the scaling of gate
depth in relation to the number of qubits. Many of the remaining research
papers that were not used were due to a lack of clearly defined circuit shape,
e.g., by only specifying resources as exponential or not.

Of the 58 algorithms considered, 27 algorithms were designed with NISQ
computers in mind [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31,
32, 33, 34], while 31 remain applicable only for larger or more universal
fault-tolerant quantum computers [35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45,
46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61]. A breakdown
of the number of algorithms by application area is given in Table 2.

###  2.1 Assumptions During Data Collection

Data collection required a number of assumptions. First, several research
papers provided resource estimates that could be used to approximate the
scaling of gate depth, but did not clearly or explicitly specify the scaling
of qubits. In these cases, qubits were assumed to scale linearly.

Second, the estimated scaling of gate depth for many of the algorithms
depended on other variables in addition to simple qubit count. For instance,
on the condition number of a specific matrix or function that is being
computed, or the desired error tolerance in a quantum simulation. In these
situations, these additional variables were treated as constants with the
assumption that their exact values would be determined when applied to a
specific problem.

Finally, we found that some of the resource estimation papers provided
specific estimates for the scaling of gate depth, while others provided
estimates for the scaling of similar (but different) variables such as gate
count, operations, runtime, or time complexity (See Table 3 for a breakdown of
the algroithms by depth estimation type). Estimates for the scaling of gate
depth, runtime, and time complexity typically account for the ability to
implement quantum gates in parallel whereas estimates for the scaling of gate
count and operations typically do not. Therefore, estimates associated with
gate count or operations were considered overestimations that needed to be
accounted for when analyzing the data.

Circuit Depth Estimate Type | Count
---|---
Gate Depth | 14
Gate Count/Operations | 19
Runtime/Time Complexity/etc. | 25
Table 3: Count of how many algorithms fall within each category of gate depth
estimation types.

###  2.2 Categorization of Circuit Shapes

Figure 2: The Quantum Volumetric Classes used in this work, as well as the
initial assumptions about which circuit depths from Equation 3 fall within
each class.

Data analysis first required identifying the circuit shapes that align with
each algorithm’s specific scaling of gate depth. We considered each
algorithm’s circuit shape, $\mathcal{S}$, to be of the form

$\mathcal{S}=n\times d(n),$ (2)

where the circuit depth $d(n)$ is explicitly some function of the number of
required qubits $n$. The scaling of circuit depth for all algorithms fell into
one of the following forms, arranged from slower-growing to faster-growing:

$d(n)\sim\begin{cases}\mathcal{O}(1)\\\
\mathcal{O}(\operatorname{polylog}(n))\\\
\mathcal{O}(\sqrt{n}\operatorname{polylog}(n))\\\
\mathcal{O}(n\operatorname{polylog}(n))\\\ \mathcal{O}(n)\\\
\mathcal{O}(n\operatorname{polylog}(n))\\\ \mathcal{O}(n^{2})\\\
\mathcal{O}(n^{2}\operatorname{polylog}(n))\\\ \mathcal{O}(n^{3})\\\
\mathcal{O}(n^{3}\log(n))\\\ \text{or}\\\ \mathcal{O}(n^{5})\end{cases}.$ (3)

Ignoring $\operatorname{polylog}(n)$ factors in Equation 3, we filtered the
possible circuit shapes into the following “Quantum Volumetric Classes” listed
in Figure 2. Most resource estimates for scaling are rough order of magnitude
estimates and ignore things such as overheads which scale at least of order
$\log(n)$ [62]. Therefore, we end up with a more reasonable number of
categories with minimal loss of informational value.

Mathematically, these classes can be written as

$\begin{split}\text{QV-1}&=\max_{n<n_{\text{max}}}\Big{(}\min\big{[}n,d\big{]}\Big{)}\\\
\text{QV-2}&=\max_{n<n_{\text{max}}}\Big{(}\min\big{[}n,d^{1/2}\big{]}\Big{)}\\\
\text{QV-3}&=\max_{n<n_{\text{max}}}\Big{(}\min\big{[}n,d^{1/3}\big{]}\Big{)}\\\
\text{QV-4}&=\max_{n<n_{\text{max}}}\Big{(}\min\big{[}n,d^{1/4}\big{]}\Big{)},\end{split}$
(4)

where the variables are defined just as they were in Equation 1. It should be
noted that we do not follow the exponential convention from the original
quantum volume for our classes. In particular, although QV-1 contains the
exact same information as the original quantum volume $V_{Q}$, these two are
not equal. Instead the relationship between the two is given as

$\text{QV-1}\equiv\log_{2}(V_{Q}).$ (5)

This should not present an issue as it is already common to quote
$\log_{2}(V_{Q})$ rather than $V_{Q}$ anyways.

As an example of how these classes work, a given quantum computer may score a
value of $n$ for the QV-1 metric. This corresponds to the determination that
the device is able to implement square circuits up to a size
$\mathcal{S}=n\times n,$ and is equivalent in this case to a quantum volume of
$\log_{2}(Q_{V})$. Likewise a device with a QV-2 score of $n$ means the device
can successfully implement circuits whose depth grows quadratically with qubit
number up to a circuit shape of $\mathcal{S}=n\times n^{2}$. In general, a
value of $n$ for the QV-$k^{\text{th}}$ volumetric metric corresponds to the
circuit of shape $\mathcal{S}=n\times n^{k}.$

The following subsections walk through the data analysis process using these
classes of quantum volume metrics. First we will present initial results
without any adjustments to the data. Then, we will highlight adjustments made
to the data based on (1) an assumption that algorithms will require additional
gate depth to account for gate overhead and (2) the lack of parallelism in
resource estimates for the scaling of gate count and operations. The final
subsection then presents final results.

##  3 Results

The following subsections walk through the data analysis process using these
classes of quantum volume metrics. In subsection 3.1, we will present initial
results without any adjustments to the data. Then, subsection 3.2 will
highlight adjustments made to the data based on (1) an assumption that
algorithms will require additional gate depth to account for gate overhead and
(2) the lack of parallelism in resource estimates for the scaling of gate
count and operations. Section 3.3 then presents final results.

###  3.1 Initial Results

Class | Circuit Depth | Counts
---|---|---
QV-1 | $\mathcal{O}(n)$ | 16 (28%)
$\mathcal{O}(\log(n))$ | 8 (14%)
$\mathcal{O}(\log^{x}(n))$ | 6 (10%)
$\mathcal{O}(1)$ | 2 (3%)
$\mathcal{O}(\sqrt{n}\operatorname{polylog}(n))$ | 1 (2%)
QV-2 | $\mathcal{O}(n^{2})$ | 10 (17%)
$\mathcal{O}(n\operatorname{polylog}(n))$ | 2 (3%)
QV-3 | $\mathcal{O}(n^{3})$ | 8 (14%)
$\mathcal{O}(n^{2}\operatorname{polylog}(n))$ | 1 (2%)
QV-4 | $\mathcal{O}(n^{3}\log(n))$ | 2 (3%)
QV-5 | $\mathcal{O}(n^{5})$ | 2 (3%)
Table 4: Number of algorithms by circuit depth scaling and grouped by QV
class.

Our initial sorting of algorithms into QV classes is shown in Figure 2. The
counts for the number of algorithms for each of the given depth scalings of
Equation 3 is given in Table 4, together with the initial sorting by QV class.
Under this initial grouping, 33 (57%) algorithms fall into QV-1, 12 (21%) into
QV-2, 9 (16%) into QV-3, and 2 (3%) into either QV-4 or QV-5. Therefore the
original quantum volume, represented by our QV-1 class, accounts for only
$57\%$ of the quantum algorithms surveyed. If we add the additional categories
of QV-2, QV-3, and QV-4 to the original quantum volume metric, we would
increase coverage to $93\%$ of algorithms surveyed.

Figure 3: Initial breakdown of application areas by QV class.

When analyzing the initial data by application area we notice a few trends
(see Figure 3). First, machine learning and optimization algorithms primarily
seem to align with QV-1. Many-body physics and chemistry algorithms align
evenly across QV-1 and the combination of QV-2 and QV-3. Quantum data hiding
algorithms align with QV-1 for watermarking and QV-2/QV-3 for steganography.
Numerical solvers (i.e., Shor’s algorithm) align with QV-3 or QV-4.

###  3.2 Adjustments to Results

After an initial analysis, the categorization of algorithms into their
respective QV classes was adjusted to account for the following assumptions.
First, it is anticipated that implementing algorithms on quantum computers
will require gate overhead due to the physical limitations of qubit
connectivity within the physical devices. Qubits will likely not be able to
directly interact with all other qubits; therefore, additional quantum gates
will likely be needed to make the necessary connections. This gate overhead is
expected to be at least of order $\log(n)$ [62].

Figure 4: Adjustments to the categorization of algorithms into QV classes as
origionally represented in Figure 2 for different scaling behaviors for the
circuit depths.

Keeping this overhead in mind, the initial data presented in subsection 3.1 is
likely an underestimation of actual gate depth requirements. To account for
this, algorithms with circuit depth estimates scaling exactly with the shape
of their respective QV metrics are pushing the upper limits of the metric, and
they should therefore move to the next higher QV metric to leave room for
potential gate overhead. To adjust for this, algorithms with circuit depths
$\mathcal{O}(n)$ were moved from QV-1 to QV-2, $\mathcal{O}(n^{2})$ were moved
into QV-3, and $\mathcal{O}(n^{3})$ were considered part of QV-4. This change
is represented graphically in Figure 4.

Depth | Adjust Class | Keep Initial Class
---|---|---
$\mathcal{O}(n)$ | 12 | 4
$\mathcal{O}(n^{2})$ | 6 | 4
$\mathcal{O}(n^{3})$ | 4 | 4
Table 5: A count of how many algorithms for borderline cases were adjusted to
the next higher QV class according to method shown in Figure 4. This
assessment was based on which circuit depth estimation type was used (see
Table 3). Estimations based on total gate counts were kept at their initial QV
class, while all others were adjusted.

A second consideration was the fact that the resource estimates of the various
algorithms were not all of the same type (see Table 3). In particular,
resource estimations that rely on assumptions of total gate or operation
counts do not take into account parallelisation of the actual algorithm. Since
these cases already represent overestimations of gate depth, we did not move
any such algorithm to a higher QV class. Counts for how many algorithms were
moved to a higher QV classes is shown in Table 5.

###  3.3 Final Results

Class | Circuit Depth | Counts
---|---|---
QV-1 | $\mathcal{O}(n)$ | 4 (7%)
$\mathcal{O}(\log(n))$ | 8 (14%)
$\mathcal{O}(\log^{x}(n))$ | 6 (10%)
$\mathcal{O}(1)$ | 2 (3%)
$\mathcal{O}(\sqrt{n}\operatorname{polylog}(n))$ | 1 (2%)
QV-2 | $\mathcal{O}(n)$ | 12 (21%)
$\mathcal{O}(n^{2})$ | 4 (7%)
$\mathcal{O}(n\operatorname{polylog}(n))$ | 2 (3%)
QV-3 | $\mathcal{O}(n^{2})$ | 6 (10%)
$\mathcal{O}(n^{3})$ | 4 (7%)
$\mathcal{O}(n^{2}\operatorname{polylog}(n))$ | 1 (2%)
QV-4 | $\mathcal{O}(n^{3})$ | 4 (7%)
$\mathcal{O}(n^{3}\log(n))$ | 2 (3%)
QV-5 | $\mathcal{O}(n^{5})$ | 2 (3%)
Table 6: Number of algorithms by circuit depth scaling and grouped by QV class
based on the adjustments described in subsection 3.2.

After adjusting the data as described in subsection 3.2 we arrived at the
final results, presented in Table 6 below. Under this adjusted grouping, the
utility of QV-1 is reduced, as it now only accounts for 21 (36%) of the
algorithms, while QV-2 saw a significant boost to 18 (31%) algorithms. QV-3
increased slightly to 11 (19%), while QV-4 now has 6 (10%) of all algorithms
surveyed. QV-5 remained at 2 (3%). Therefore we see, a set of metrics
consisting of QV-1, QV-2, QV-3, and QV-4 should cover a vast majority of
quantum algorithms ($91\%$).

We again analyzed the updated data by application area, which we represent
graphically in Figure 5 Machine learning and optimization algorithms continue
to align most strongly with QV-1. Many-body physics and chemistry algorithms
now align most strongly with QV-2. Although machine learning, optimization,
and many-body physics and chemistry algorithms align most strongly to a
specific QV metric, they each have at least one implementation for each QV
metric. Quantum data hiding algorithms now align with QV-2 for watermarking
and QV-3/QV-4 for steganography. Numerical solvers (i.e. Shor’s algorithm)
continue to align with QV-3 or QV-4.

Figure 5: Adjusted breakdown of application areas by QV class.

##  4 Conclusions

Quantum computing is a rapidly developing technology that promises to be
broadly disruptive across many fields. This necessitates the development of
better metrics to enable potential end users to evaluate and compare hardware
platforms for their desired applications, as well as to track the development
of the technology over time. The generalized quantum volume classes presented
in this work (see Equation 4) provide a generalization to the original quantum
volume benchmark that is applicable to a much broader range of potential
applications, while not becoming too complex to be useful to the potential end
users for these applications.

Adoption of these metrics by the community would help in a number of ways.
These metrics more explicitly tie the power of quantum computing to specific
use cases, enhancing the utility to end-users. Focusing on developing metrics
from the end-user perspective will also help companies, governments, and other
entities to make more informed decisions concerning this field. Adoption of
these metrics or similar frameworks will help guide the field by quantifying
how success and improvement is measured. Having a consistent framework should
also help minimize some of the assumptions specified in section 2.

In the coming years we can expect to see more powerful quantum computing
systems with both more qubits and better fidelities. Soon, we will even see
systems whose error rates fall below error correction thresholds, opening the
door to further progress through the application of quantum error correction
to these systems. The original quantum volume does not specify how to account
for error correction, i.e., whether the specified square circuit shape applies
at the logical or physical level. In order for a metric to be sufficiently
simple and useful to end users who may not be experts in quantum computing,
the metric should be defined at the logical level to best align with the
application space. However, it would also be useful to be able to connect the
user space metric, such as that presented here, with estimates of system level
performance. Such an accounting is beyond the scope of this work, but is
forthcoming in a future paper.

## Acknowledgements

The Homeland Security Act of 2002 (Section 305 of PL 107-296, as codified in 6
U.S.C. 185), herein referred to as the “Act,” authorizes the Secretary of the
Department of Homeland Security (DHS), acting through the Under Secretary for
Science and Technology, to establish one or more federally funded research and
development centers (FFRDCs) to provide independent analysis of homeland
security issues. MITRE Corp. operates the Homeland Security Systems
Engineering and Development Institute (HSSEDI) as an FFRDC for DHS under
contract 70RSAT20D00000001.

The HSSEDI FFRDC provides the government with the necessary systems
engineering and development expertise to conduct complex acquisition planning
and development; concept exploration, experimentation and evaluation;
information technology, communications and cyber security processes,
standards, methodologies and protocols; systems architecture and integration;
quality and performance review, best practices and performance measures and
metrics; and, independent test and evaluation activities. The HSSEDI FFRDC
also works with and supports other federal, state, local, tribal, public and
private sector organizations that make up the homeland security enterprise.
The HSSEDI FFRDC’s research is undertaken by mutual consent with DHS and is
organized as a set of discrete tasks. This report presents the results of
research and analysis conducted under: 70RSAT22FR0000021, “DHS Science and
Technology Directorate TCD QIS Capabilities.”

The results presented in this report do not necessarily reflect official DHS
opinion or policy.

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Nuclear modification factors of strange mesons measured by PHENIX

A. Berdnikov1, Ya. Berdnikov1, D. Kotov1, D. Larionova1, Iu. Mitrankov1, M.
Mitrankova1 and V. Borisov1*

1 Peter the Great Saint-Petersburg Polytechnic University, St.Petersburg,
Russia

*<EMAIL_ADDRESS>
50th International Symposium on Multiparticle Dynamics
(ISMD2021)
12-16 July 2021 10.21468/SciPostPhysProc.?

## Abstract

The question of the existence and properties of the quark-gluon plasma (QGP)
nowadays are the subject of detailed systematic study. Particles that contain
strange quarks can be considered as a great tool to study flavor dependence of
the parton energy loss in the QGP and strangeness enhancement. In this paper,
we present the most recent PHENIX results on nuclear modification factors of
$K^{\pm}$, $K^{*0}$, $\phi$ mesons as a function of $p_{T}$ and the number of
participants in $p$+$p$, $p$+Al, 3He+Au, Cu+Cu, Cu+Au, Au+Au, and U+U
collisions at top RHIC energies. The light hadron production at high-$p_{T}$
is apparently independent on quark content. The coalescence mechanism might be
an answer for strangeness and baryon enhancement at moderate $p_{T}$.

###### Contents

  1. 1 Introduction
  2. 2 $R_{AB}$ results in small systems collision
  3. 3 $R_{AB}$ results in heavy ion collisions
  4. 4 Conclusions

##  1 Introduction

For the decades one of the most important goals of the high energy physics was
investigation of matter which filled the early Universe, so-called quark-gluon
plasma (QGP) [1]. In laboratory conditions the QGP can be studied with
relativistic ion collisions. The strangeness production was originally
proposed as a signature of QGP formation [2] and still remains experimentally
popular observable since strange hadrons are produced abundantly in the QGP
and can be measured over a large kinematic domain. The comparison of the
(hidden)strange particles production to the production of hadrons that contain
only first-generation quarks is a good tool to study such QGP effects as
strangeness enhancement [2], recombination [3] and radial flow [4] in
intermediate transverse momentum ($p_{T}$) range and the flavor dependence of
parton energy loss [5] in high $p_{T}$ range.

Nowadays it is generally accepted that the QGP is formed in the relativistic
heavy ion collisions such as Cu+Cu, Cu+Au, Au+Au at $\sqrt{s_{{}_{NN}}}$ = 200
GeV, and U+U at $\sqrt{s_{{}_{NN}}}$ = 193 GeV [6, 7]. Measurements of
strangeness production in large collision systems provide an opportunity to
enrich and expand the understanding of the QGP properties.

However, such QGP characteristic as minimal conditions sufficient for its
formation is still under investigation in a broad set of geometry controlled
small collision systems. Recent PHENIX results indicate that the energy
density in $p$/$d$/3He + Au collisions at $\sqrt{s_{{}_{NN}}}$ = 200 GeV is
ample for the creation of the deconfined state of matter in contrast to $p$+Al
collisions at the same energy [8]. Minimal condition to form the QGP might be
in between $p$ \+ Al and $p$ \+ Au collisions at $\sqrt{s_{{}_{NN}}}$ = 200
GeV. However, the signatures of the QGP were observed in $p$ \+ Al collisions
at $\sqrt{s_{{}_{NN}}}$ = 200 GeV at forward and backward rapidities [9].
Thus, the experimental determination of the QCD phase transition critical
point require further scrutiny.

This paper presents the study of strangeness production from small to large
collision systems measured by PHENIX [10] at top RHIC [11] energies. The
nuclear modification factors of $K^{\pm}$, $K^{*0}$, $\phi$ mesons were
measured as a function of $p_{T}$ and the number of binary nucleon-nucleon
collisions in $p$+$p$, $p$+Al, 3He+Au, Cu+Cu, Cu+Au, Au+Au, and U+U
collisions. The obtained quantities were compared to previous non-strange
hadron production results to reveal the QGP effects.

##  2 $R_{AB}$ results in small systems collision

Figure 1: The nuclear modification factors of $\pi^{\pm}$, $\pi^{0}$,
$K^{\pm}$, $K^{*0}$, $p$($\bar{p}$), and $\phi$ for the most central (0-20%)
(a, c) and the most peripheral (40-72%) (b) (60-88%) (d) $p$+Al and 3He+Au
collisions at $\sqrt{s_{{}_{NN}}}$ = 200 GeV. Here and below error bars and
rectangles correspond to statistical and systematic uncertainties
respectively.

Medium-induced effects on light hadron production are studied using the
nuclear modification factor [12]:

$R_{AB}=\frac{d^{2}N_{AB}/dydp_{T}}{N_{coll}\cdot d^{2}N_{pp}/dydp_{T}}$

where $d^{2}N_{AB}/dydp_{T}$ ($d^{2}N_{pp}/dydp_{T}$) is the per-event yield
of particle production in $A+B$ ($p+p$) collisions and $N_{coll}$ is a number
of binary nucleon-nucleon collisions.

The comparison of the nuclear modification factors measured for different
light hadrons ($\pi^{\pm}$, $\pi^{0}$, $K^{\pm}$, $K^{*0}$, $p$($\bar{p}$),
and $\phi$) are shown in the Fig. 1 for the most central (0-20% centrality
class [13]) and the most peripheral $p$ \+ Al (40-72%) and 3He+Au (60-88%)
collisions at $\sqrt{s_{{}_{NN}}}$ = 200 GeV. In central 3He+Au collisions
protons yields are enhanced relatively to the binary scaled yields in $p$ \+
$p$ collisions, while all mesons $R_{AB}$ independently of quark content lie
on the same curve. However, all light hadron production show conformity in $p$
\+ Al collisions. This result might draw an assumption that coalescence
mechanism [3] plays a crucial role for light hadron production 3He+Au
collisions at $\sqrt{s_{{}_{NN}}}$ = 200 GeV at midrapidity ($\eta$ < 0.35),
whereas its contribution to hadronization in $p$ \+ Al collisions is
insignificant. In the most peripheral collisions in both systems there is no
significant modification of all light hadron production.

##  3 $R_{AB}$ results in heavy ion collisions

In the Fig. 2 $R_{AB}$ of $K^{*0}$, $\phi$ mesons and charged hadrons in
comparison to recently published $\pi^{0}$ and $\eta$ results [14, 15] in the
most central (0-20%) Cu+Au and U+U collisions are presented. The data shows an
ordering at intermediate $p_{T}$: protons $R_{AB}$ are larger than $\phi$ and
$K^{*0}$ mesons (containing strange quarks) $R_{AB}$ which are larger than
light flavored mesons $R_{AB}$. These features of hadron production can be
explained with an interplay of strangeness enhancement and recombination model
[3]. At high $p_{T}$ a flavor independent suppression of hadron production is
observed. This result may be the consequence of the parton energy loss in the
QGP.

In the Fig. 3 integrated $\langle R_{AB}\rangle$ of $\phi$ meson in
intermediate $p_{T}$ range are presented. The strange mesons production in
heavy ion collisions scales with a number of binary nucleon-nucleon collisions
and seems to depend on nuclear overlap size (represented by $N_{coll}$), but
not on the initial geometry of the collision system.

Figure 2: The nuclear modification factors of light hadrons for the most
central (0-20%) Cu + Au collisions at $\sqrt{s_{{}_{NN}}}$ = 200 GeV (a) and U
+ U collisions at $\sqrt{s_{{}_{NN}}}$ = 193 GeV (b).  Figure 3: The
integrated nuclear modification factor as a function of $N_{coll}$ for $\phi$
mesons in Cu+Cu, Cu+Au and Au+Au collisions at the energy of
$\sqrt{s_{{}_{NN}}}$ = 200 GeV and U+U collisions at the energy of
$\sqrt{s_{{}_{NN}}}$ = 193 GeV. The tilted error bars represent the anti-
correlated uncertainty on the y and x-axis due to the $N_{coll}$ calculations.

##  4 Conclusions

The nuclear modification factors of $K^{\pm}$, $K^{*0}$, and $\phi$ mesons
were measured in p+Al, He+Au, Cu+Cu, Cu+Au, Au+Au collisions at
$\sqrt{s_{{}_{NN}}}$ = 200 GeV, and U+U collisions $\sqrt{s_{{}_{NN}}}$ = 193
GeV at midrapidity as a function of $p_{T}$ and $N_{coll}$. The results were
compared to non-strange mesons and (anti)protons $R_{AB}$.

In central 3He + Au collisions proton production is enhanced compared to
binary scaled yields in $p$ \+ $p$ collisions. Whereas no strangeness
enhancement is observed within uncertainties. The most central $p$ \+ Al
collisions exhibit minimal or no apparent QGP effects. Obtained results
contribute to the concept that coalescence plays the predominant role in 3He +
Au collisions, but the energy density is insufficient for observation of
strangeness enhancement. In $p$ \+ Al collisions there seems to be no evidence
of recombination at midrapidity.

In heavy ion (Cu + Au and U + U) collisions no flavor dependence of light
hadron production at high-$p_{T}$ is observed. An obtained $R_{AB}$ ordering
($p$($\bar{p}$) $R_{AB}$ are larger than $\phi$ and $K^{*0}$ mesons $R_{AB}$
which are larger then light flavored mesons $R_{AB}$) at moderate $p_{T}$ in
central collisions suggests that coalescence might be an answer for
strangeness and baryon enhancement. The production of the strange mesons in
heavy ion collisions scales with number of collisions and seems to depend on
nuclear overlap size, but not on its geometry.

## Acknowledgments

We acknowledge support from Russian Ministry of Education and Science, state
assignment for fundamental research (code FSEG-2020-0024) in the $\phi$ meson
part of the analysis.

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