Understanding quantum bit notation

Turns out it’s somewhat intuitive!

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Photo by Mark Garlick on Science Photo Library
  • After a qubit collapses to |0or |1, it does not go back into superposition. If it collapsed to |0, it is now |0.

Expressing 1 qubit

Expanded notation for 1 qubit

Let’s say we have a qubit, |φ. Perhaps you have seen expanded notation:

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Expanded notation. (Fig 1)
  • b² is the probability that |φ will collapse to |1when measured
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Example #1. Note that the |φ₁‘phi one’ (left) is a variable representing the qubit. (Fig 2)
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Example #2. (Fig 3)
  • By similar logic, |0represents a qubit that always collapses to (or “has collapsed to”) |0.
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(Fig 4)

Vector notation for 1 qubit

There exists another way to express qubits, which involves vectors. If you’ve never seen vectors or matrices, you should check out this introduction.

Definition of |0⟩ and |1⟩ in vector notation
Definition of |0⟩ and |1⟩ in vector notation
(Fig 5)
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(Fig 6)
Vector notation
Vector notation
Vector notation. (Fig 7)
  • b² is still the probability that the qubit will collapse to |1 when measured
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Example #3. This is the vector notation of Example #1. (Fig 8)
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Example #4. This is the vector notation of Example #2. (Fig 9)
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Example #4: |φ₄has a 100% chance of collapsing to |1 when measured, so it is |1. (Fig 10)
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Example #5. (Fig 11)
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Note, it just happens that (1)² = 100% and (0)² = 0%, so it seems the values weren’t squared. (Fig 12)
  • (bottom number)² = probability the qubit collapses to |1when measured

Expressing 2+ qubits

Expanded notation for 2+ qubits

We can use expanded notation in a very similar manner. Note, though, that each additional qubit doubles the number of fundamental states:

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1 qubit, 2 states. (Fig 13)
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2 qubits, 4 states. (Fig 14)
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3 qubits, 8 states. (Fig 15)
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Expanded notation of a 4-qubit system. (Fig 16)

Vector notation for 2+ qubits

A multi-qubit system can also be represented as a vector. For simplicity, let’s consider a 2-qubit system (we write |00to |11in ascending binary order):

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Example #6: φ₆ is made of 2 qubits. (Fig 17)
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Example #6. (Fig 18)
  • qubit 2 (blue) in state |1
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(Fig 19)
  • y² = probability that qubit 1 collapses to |1 when measured
  • z² = probability that qubit 2 collapses to |0 when measured
  • w² = probability that qubit 2 collapses to |1 when measured
  1. It is also x² z² since collapsing to |00 means:
  • qubit 2 collapses to |0(with probability z²)
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(Fig 20)
  • x² w² (qubit 1 collapses to |0and qubit 2 collapses to |1)
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(Fig 21)
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(Fig 22)
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(Fig 23)
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(Fig 24)
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Vector notation for 2 qubits. (Fig 25)
  • (3) compute the tensor product
  • (4) rewrite in terms of a, b, c, d using the equations from Fig 24
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(Fig 26)
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Notice that there is no |01in the vector representation because it is implied. (Fig 27)
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(Fig 28)
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Remember, the values are squared to find the coefficients. (Fig 29)
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(Fig 30)
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(Fig 31)

What next?

Understanding qubits in vector notation becomes very helpful when you begin to work with quantum logic gates. The gates can be represented as matrices so that their interactions with qubits become matrix multiplications. If you’re interested in how that works, check out this lecture about the mathematical representation of quantum computing.

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