ZX Calculus - Another Perspective on Quantum Circuits. Part I
Recently, stumbled across a tensor network-type framework which was completely new to me - the ZX Calculus. The ZX Calculus is not only a neat way of representing possibly complicated mathematical equations, it also gives explicit rules to alter and simplify those expressions. The ZX Calculus is particularly suited to describe matters in quantum information, which is why I'd like to provide a neat example of how to use this framework. As you might already know, quantum circuits can be fully analysed and understood with the help of tensor networks (actually, they are tensor networks) [1]. However, the ZX Calculus is a specific framework which gives a very illustrative graphical way of understanding quantum circuits, while the typical tensor network approaches are mostly tailored for many body problems.
All of the following is taken from [2], a very comprehensive introduction to the ZX Calculus and I fully recommend to go through this paper if the following glimpse into the topic made you curious.
In the following we will set up the very basic set of definitions and rules in order to understand how to evaluate the outcome of the well-known Bell circuit which creates a maximally entangled Bell state:
Basic Definitions: Spiders and Vectors
The most fundamental definition in the ZX Calculus is the spider. The Z-Spider has n inputs and m outputs and is defined as follows:
Thus, such a spider is simply a way of representing a specific kind of 2^n x 2^m matrices. Here, the |0> and |1> denote the basis states of the Pauli Z operator. Similarily, an X-Spider can be defined in terms of another basis, the eigenstates of the Pauli X operator, |+> and |- >:
Thus, the color of the dot encodes information about the basis. The usage of the basis states of both Pauli X and Pauli Z is eponymous for the ZX Calculus. One could have chosen the Pauli Y basis as well, however the choice of X and Z results in nice symmetry properties [2, p.22].
From this, we can already conclude the first identity which we will need to evaluate the Bell circuit: Set n=m=1 as well as α=0. With these parameters, the spides become plain 2x2 identity matrices (just look at the definitions!). While α=0 is denoted with an empty dot, this observation can be represented as:
Thus, as soon as we encounter single, plain dots with one incoming and one outgoing leg, we can remove them.
Additionally, we will need to know, how to represent simple basis vectors in this diagrammatic language. This is simply done by using dots with a single leg and the following simple consideration according to the definitions of the spiders:
Of course, one can also describe |- > and |1> states, just apply α=π respectively. Note that we omit global phases here; thus using a simple equality sign is actually a delicate matter.
The Hadamard Gate
The Hadamard gate is a unitary gate which simply transforms between the X and Z basis; e.g. applying the Hadamard gate to a |0> state will result in |+>. Its graphical representation is just a plain box with one outcoming and one incoming leg - its action on the basis vectors is as follows:
Actually, this is one special case of the more general rule, that the application of Hadamards changes colors as follows:
This of course also holds if the colors are inversed. In general, all ZX rules hold under coherent exchange of colors.
The CNOT Gate
Another central gate in quantum computing is the CNOT gate, which is a controlled NOT gate, i.e. the target qubit is only flipped if the control qubit is |1>, otherwise nothing happens. This 2-qubit-gate can be represented as
The equality sign should be taken with care as well, because the left is in the quantum circuit notation, while the right is in ZX calculus notation. Its construction is explicitly explained in [2, pp. 11]. Since it is a bit lengthy to go through it by representing the diagrams as matrices, I leave it to you to check it in the reference in case you are interested.
The Fusion Rule
In general, it is possible to "fuse" dots of the same color, while adding their phases. Note that it is addition mod 2π because α and β are the exponents of e.
Later, we will only use a special case of this, namely that we can fuse dots of the same color which are connected by one line.
Now, we have finally settled the rough framework for analyzing the Bell circuit, which will do in the next part!
---
References:
The ZX graphics were created with tikzit.github.io. Furthermore, you can find a lot of valuable information on zxcalculus.com.
[1] Tensor Networks in a Nutshell - Biamonte, Bergholm. 2017. arXiv:1708.00006
[2] ZX-calculus for the working quantum computer scientist - Wetering. 2020. arXiv:2012.13966
47 notes
·
View notes
Quantum Revolution's Use By The NQISRCs.
The NQISRCs are all committed to the promotion of quantum information science, despite having different specialties and resources.
The behavior of nature at the tiniest scales is being used by five National Quantum Information Science Research Centers (NQISRCs) to create technology for the most challenging issues in science.
The NQISRCs have supported the DOE's goal to advance the country's energy, economy, and national security since 2020 with funding from the Office of Science under the U.S. Department of Energy (DOE). The centers establish a vibrant environment for quantum innovation and co-design by creating a national quantum ecosystem and workforce made up of researchers from over 70 different universities throughout the United States.
The NQISRCs combine cutting-edge DOE facilities, top expertise at national labs and American colleges, and the innovative spirit of American technology businesses.
Because of this, the centers are expanding the realm of what is feasible in terms of quantum computers, sensors, devices, materials, and much more.
A DOE national laboratory oversees each national center:
Brookhaven National Laboratory is the Center for Quantum Advantage (C2QAprincipal )'s designer.
Argonne National Laboratory's Q-NEXT initiative
Leading the Quantum Science Center is Oak Ridge National Laboratory.
Lawrence Berkeley National Laboratory's Quantum Systems Accelerator (QSA)
Fermi National Accelerator Laboratory's Superconducting Quantum Materials and Systems Center (SQMS) is in charge of this project.
According to Q-NEXT Director David Awschalom, "Each center is a strong force for quantum information science on its own, pushing the boundaries of computers, physics, chemistry, and materials science to deliver transformative new technologies to the country."
But when they work together, they form a powerful national force that gives quantum science and engineering a unique place in the United States and positions it to lead the world in this area.
Quantum information science (QIS), a fast developing branch of study, looks at the quantum aspects of nature to provide new, potent methods to process information in fields as diverse as health, energy, and finance.
Researchers might create new, very precise sensors, powerful computers, and safe communication networks by modifying the most basic properties of matter.
In order to do this, the institutes are developing prototype quantum computers and sensors and evaluating their influence and performance on different technology platforms and architectural designs.
According to C2QA Director Andrew Houck, "there are numerous options and possibilities to be made in the development of quantum computing, and knowing how present devices fail indicates to us the road ahead."
Despite significant advancements in the area, the NQISRCs are able to complete this unexpectedly difficult job since contemporary quantum computers are still too noisy and error-prone to do effective calculations.
In order to get over these noise restrictions and create devices with a quantum advantage, it is essential to understand the quantum behavior of materials.
The national labs are in a unique position to provide cutting-edge resources and expertise that direct the comprehension and removal of these constraints.
According to SQMS Director Anna Grasselino, "DOE has invested for years in state-of-the-art technology, techniques, and facilities at national laboratories, which provide unique prospects to allow a jump in performance of quantum devices."
Especially since QIS can promote our aim of comprehending the universe at its most basic level, we are thrilled to provide world-leading knowledge to achieve revolutionary improvements in QIS.
To provide the groundwork for future scientific breakthroughs, the multidisciplinary teams at the NQISRCs jointly build quantum technologies.
New materials and potent quantum sensors that, when paired with medical imagers, might detect tissue at the individual-cell level and provide far higher sensitivity to today's magnetic resonance imaging equipment are just two examples of how advances in QIS can benefit society as a whole.
The co-design effort across the NQISRCs might result in quicker medication and vaccine development, innovative materials, advances in transportation and logistics, and more secure financial networks by comprehending what enables and limitations various quantum technologies and what tools need to be created.
NQISRC researchers utilize top-tier DOE Office of Science user facilities and programs as a national ecosystem, including the Advanced Photon Source at Argonne National Laboratory, the Oak Ridge Leadership Computing Facility at Oak Ridge National Laboratory, the Advanced Light Source at Lawrence Berkeley National Laboratory, the National Synchrotron Light Source II at Brookhaven National Laboratory, and the superconducting technology facilities and technologies at Fermilab.
According to QSC Director Travis Humble, "DOE has provided researchers an unparalleled chance to make significant and game-changing discoveries in QIS" via the financing of these key quantum centers.
"Based on the first two years of operation, there is every reason to assume that these centers will significantly advance QIS toward real-world innovation in the next years. The innovation chain will witness an increase in the flow of discovery research."
The market-driven technologies created by its industrial partners, such as test beds and simulation tools, may also be used by laboratory and university scientists.
Each center creates a road to the commercialization of quantum technologies and, ultimately, their release to the general public by making use of these networks.
A quantum workforce will be necessary to sustain the NQISRCs' co-design initiatives for science and technology in the long run. Through institutional degree programs, joint training initiatives with business, and retraining certificate programs, all national centers are dedicated to developing a workforce with an emphasis on diversity, equality, and inclusion. This opens the door for the investigation of several more breakthroughs and important scientific issues.
QSA Director Irfan Siddiqi said, "The centers have adopted a multifaceted strategy to teach the next generation of QIS scientists and researchers and to build new pipelines for underrepresented groups." "We're all making extra steps to encourage a diverse quantum workforce in a subject that is rapidly expanding."
A platform for discussing QIS themes with high schools, college students, postdocs, and professionals was the second annual quantum summer school hosted in May by the QSC at Oak Ridge National Laboratory, for instance.
The occasion included networking opportunities for employees and students as well as seminars on workforce development methods with top business leaders.
For undergraduate students, C2QA recently launched its second six-week quantum computing summer school, QIS101, with an emphasis on developing foundational and practical skills and expanding a diverse quantum workforce.
In the summer session of C2QA, 40% of participants were female, and almost 44% came from groups that are underrepresented in QIS.
Through the Open Quantum Initiative, a group headed by the Chicago Quantum Exchange, Q-NEXT collaborates with other quantum institutions to create a more diverse and inclusive quantum workforce.
The development of an undergraduate fellowship program for underrepresented, racially underrepresented quantum physicists was one such initiative. The fellowship participants collaborated on difficult QIS projects this summer with researchers from partner universities.
QSA works closely with various regional and worldwide firms with good histories in diversity and inclusion initiatives. Additionally, it is collaborating with regional economic development organizations that are already working in the sector to create internship and apprenticeship programs and quicken starting times.
Numerous entrepreneurs, investors, CEOs, top scientists, and engineers from all around the globe participated in QSA's industry and investor roundtable events in 2021.
The inaugural Carolyn B. Parker Fellowship, named after Carolyn Beatrice Parker, the first African American woman to get a Ph.D. in physics, has been awarded, according to SQMS. The SQMS QIS Summer School, co-hosted by the Galileo Galilei Institute in Florence, Italy, and the second annual undergraduate internship program are both coming to an end while the center continues its search for a second Parker fellow.
The second NQISRC Virtual Quantum Career Fair will take place on September 14 at Brookhaven National Laboratory.
The purpose of the event is to increase awareness among the undergraduate, graduate, and postdoctoral communities of the DOE Office of Science's NQISRCs and the many QIS occupations available at the centers, including both technical and non-STEM vocations. Nearly 400 people attended the first NQISRC job fair in the autumn of 2021, with 12% coming from organizations that serve minorities.
Quantum information science advancements have the power to transform both science and society. By creating technologies that go beyond what has previously been feasible, the NQISRCs are at the vanguard of this burgeoning sector.
~ Jai Krishna Ponnappan.
Find Jai on Twitter | LinkedIn | Instagram
References And Further Reading:
Learn more about each center at DOE's website: science.osti.gov/Initiatives/QIS/QIS-Centers
0 notes