A single qubit, like a single bit, can’t accomplish much on its own. For a quantum computer to be useful, it needs to handle a large number of qubits at once. To do this, we combine multiple qubits into a single, unified mathematical object called a system.
This approach allows us to represent and manipulate dozens or even hundreds of qubits at the same time. The tool we use to build these systems is the tensor product, a mathematical operation that combines individual quantum states into a larger, more complex state.
➕ Building a System with the Tensor Product
The tensor product is a key concept in quantum computing. When we use it to combine two quantum states, say $|\psi\rangle$ and $|\phi\rangle$, we get a new state $|\psi\rangle \otimes |\phi\rangle$. This new state is a single column vector that describes both qubits together. For a system of two qubits, the resulting vector has four elements.
This isn’t just a list of the original qubits’ amplitudes; it’s a new vector whose elements are the products of every possible pairing of the original amplitudes. This process continues for larger systems, so a system of $n$ qubits is described by a single vector with $2^n$ elements.
🗺️ Reading a Quantum Circuit Diagram
Just as we create systems of qubits, we can also combine quantum gates (qugates) into larger systems. This is essential for understanding and designing complex quantum circuits. There are two primary ways to combine qugates in a circuit diagram: horizontally and vertically.
When qugates appear one after another on the same qubit line, we combine them horizontally using ordinary matrix multiplication. For example, if a qubit passes through gate A then gate B, the combined operation is represented as BA (since we read operators from right to left).
When qugates appear on different qubit lines at the same point in the circuit, we combine them vertically using the tensor product. A circuit with gate A on the top line and gate C on the bottom line is represented as $A \otimes C$. This powerful combination of matrix multiplication and tensor products allows us to analyze any quantum circuit, no matter how complex, in a clear and consistent way.
🧠 The No-Cloning Theorem
When thinking about multi-qubit systems, it’s natural to wonder if you can just copy a qubit’s state, much like you would copy a bit. But a fundamental law of quantum mechanics, known as the no-cloning theorem, forbids this. It states that you cannot perfectly clone an arbitrary quantum state.
This has profound implications for programming, as it means you can’t simply branch a quantum wire in a circuit diagram. A qubit cannot be duplicated, which means we must find new ways to approach common programming tasks. This constraint forces us to be more creative in how we design our algorithms, making quantum programming a truly unique and challenging field.
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Glassner, Andrew. Quantum Computing: From Concepts to Code. No Starch Press, 2025.
More Topics
- The Quantum Threat: How Shor’s Algorithm Puts Modern Encryption at Risk
- The Bernstein-Vazirani Algorithm: Unmasking a Secret With a Single Query
- Simon’s Algorithm: The First Quantum Speedup That Left Classical Computers in the Dust
- Cracking the Code: How Quantum Parallelism Solves Deutsch’s Problem in One Step
- The Deutsch-Jozsa Algorithm: An Exponential Leap in Quantum Problem Solving
- Your First Quantum Program: The ‘Hello, World!’ of a Qubit
- The ‘Spooky Action’: What Happens When Qubits Entangle?
