One of the most basic concepts in modern days quantum computing is a quantum circuit. It is essentially the second thing most quantum computing students learn after getting familiar with a qubit is the quantum circuit. Though the initial circuits (resembling “Hello World!” examples from other computer science fields) seem to be fairly simple, the subject is actually quite complex. In this blog post I cover the essentials of circuit notation: what it is, what are the principles and conventions used when plotting them. I also focus on the most popular quantum computing framework by far – Qiskit – for reasons that I will explain later.
In essentials, circuits are models of computation in which information is carried by wires through a network of gates, which represent operations on the information that is being carried. Quantum circuits are a specific model of computation based on this more general concept. If this definition seems to be too abstract for you, you can think of circuits as graphs where nodes are gates and edges are wires.
The concept itself comes from the field of electric engineering. Basically, classical circuits are no different from electric ones, since they follow exactly the same principals. The only difference is that computing circuits (regardless being classical or quantum) are rarely cyclic, and they usually represent a finite sequence of operations that does not permit a loop. Quantum circuits operate on completely different physical laws, but the concept itself is as described above.
Circuits can be boolean or arithmetic. Boolean circuits are used to compute logical values (true and false) for predicates, whereas arithmetic circuits are used to do maths. Fun fact, it can be also done with the boolean circuits, but is highly inefficient.
Circuit Diagrams
In the quantum circuit model, wires represent qubits. Wires are plotted as horizontal lines stretching from left to right, usually denoted with some sort of a qubit’s identifier (q for example).
Gates represent operations on these qubits. They are usually drawn as squares of different colouring with letters distinguishing their type. There are exceptions depending on what gate is used in a given circuit. The operations they represent can be a computation or a measurement.
Below is a diagram of a sample circuit.
In the diagram, you can see one qubit (q) represented with a wire (the horizontal line), and one gate (the dark blue square with X) representing a computation. In this case it is a boolean negation implemented with a NOT (X) gate. You can read more about this gate here if you are unfamiliar with it.
Quantum computations must be eventually measured in order to make some use of them. After all, if we applied only computation steps, we would never know what we actually computed.
A measurement is represented in a diagram with a gray square with a control on it. It always must be accompanied by a double horizontal line representing a classical register. The classical register in quantum computing is used to store results of the computations in a form that is understandable by classical computers that oversee QPUs (Quantum Processing Units). Let’s take a look at an example.
Here, the measurement operation is represented with the grey square with a control, and the bit is represented with a double horizontal line (annotated c). Transfer of a state between a quantum and a classical register is drawn with a double line arrow.
Circuit Depth and Width
As you may intuitively think, circuits may vary in size. In fact, there are two parameters that describe it: width and depth.
Quantum circuit width refers to the number of qubits actively involved in a quantum circuit. In other words, it determines the size of the quantum state being manipulated. For example, a circuit acting on two qubits has a width of two like in the following diagram.
Here we have two qubits (q0 and q1) that are being used simultaneously in a computation (represented with a CNOT gate).
It is important to remember that even if some qubits are idle during certain operations, their presence contributes to the overall circuit width.
The other circuit’s parameter – quantum circuit depth – refers to the number of quantum gates used at a particular step or layer of a quantum circuit. To put it more formally, it relates to the computational parallelism or the number of simultaneous operations being performed in the circuit.
Let’s see it in an example. In the following diagram, there are two gates acting on two different qubits (q0 and q1) simultaneously, so the depth of the circuit at that layer is two.
Width and depth are independent values, though depth can never exceed the overall width of the circuit. Moreover, it is important to remember that while depth is measured for a given part of the circuit, the width is a global value of all the qubits constituting the given circuit.
Qubits Ordering
I have already hinted this in the introduction. Every quantum programming framework has its own convention of drawing quantum diagrams and ordering qubits in them. I use almost exclusively Qiskit, since it is by far the most popular programming framework for QPUs, as well as having really great learning resources. Therefore, I will explain in this post its respective convention.
In Qiskit, the topmost qubit in a circuit diagram has index 0, and corresponds to the rightmost position in a tuple of qubits. The second-from-top qubit has index 1, and corresponds to the position second-from-right in a tuple, and so on, down to the bottommost qubit, which has the highest index, and corresponds to the leftmost position in a tuple.
In particular, Qiskit’s default names for the qubits in an n-qubit circuit are represented by the n-tuple (qn−1, …, q0), with q0 being the qubit on the top and qn−1 on the bottom in quantum circuit diagrams.
This convention leads to confusing situations, since it is also reflected in memory arrays that store results of computations. You can read more about it in my post about endianness in Qiskit.
Plotting a Circuit
There is one final thing that must be covered before concluding. How can you get such nice quantum diagrams? In truth, plotting a circuit is fairly easy with Jupyter notebooks and Qiskit.
First, you have to define a circuit using Qiskit’s circuit API. Then, you can call the draw method to plot the circuit. It accepts an optional argument that defines what library is used to do the plotting. One of the most popular ones is the Matplotlib, which can be chosen by passing the ”mpl” string as an argument.
Here’s an example of a simple circuit and code that plots it with the Matplotlib library.
from qiskit import QuantumCircuit, QuantumRegister, ClassicalRegister
register_qubits = 1
register_bits = 1
quantum_register = QuantumRegister(register_qubits, "q")
classical_register = ClassicalRegister(register_bits, "c")
circuit = QuantumCircuit(quantum_register, classical_register)
circuit.h(quantum_register[0])
circuit.x(quantum_register[0])
circuit.measure(quantum_register, classical_register)
circuit.draw("mpl")And the resulting diagram. Viola!
Conclusion
Circuits are models of computation in which information is carried by wires through a network of gates, which represent operations on the information carried by the wires. In quantum circuits, qubits are equivalent to wires, and operations on them to quantum gates.
The amount of qubits that are being used in a given circuit is called circuit’s width. The number of gates that are being applied parallel to one or more qubits in a given step is called circuit’s depth.
Qubit ordering depends on the computational framework you use. In Qiskit, the topmost qubit in a circuit diagram has index 0, and corresponds to the rightmost position in a tuple of qubits. The bottommost qubit, which has the highest index, and corresponds to the leftmost position in a tuple.
A picture is worth a thousand words, and the same holds true with quantum circuits. The easiest way to plot a circuit is to define it programmatically, and then use the Matplotlib library to get the drawing.
I hope that this post puts an understandable picture on the somewhat complicated concept of quantum circuits, and they will become easier to reason with.