Probability amplitudes and phase
As mentioned in What is Quantum Interference?, until the system is measured, each state in the superposition of a quantum system has an associated probability amplitude. In the case of electron spin, for example, a single electron has two associated probability amplitudes, one for each spin direction.
\( |state> = \alpha |0> + \beta |1>\)In contrast to the classical probabilities we know from everyday life, probability amplitudes are complex numbers. The probability of measuring each possible state is attained by squaring the absolute value of its probability amplitude.
\( P(|0>) = |\alpha|^2 \) \( P(|1>) = |\beta|^2 \)Examples and definitions
Since probability amplitudes are complex numbers, different amplitudes can result in the same probability upon measurement.
Each of the following probability amplitudes ensures that we measure the |1> state with certainty (100% probability):
\( |s> = 0 \times |0> + 1 |1> \) \( |s> = 0 \times |0> – 1 |1> \) \( |s> = 0 \times |0> + i |1> \) \( |s> = 0 \times |0> – i |1> \)Or, more generally:
\( |s> = 0 \times |0> + e^{i \theta}* |1> \)where theta can be any real number.
*According to Eulers formula, \(e^{i \theta} = cos(\theta) + i sin(\theta)\). The corresponding probability upon measurement for a state is \(|cos(\theta)|^2 + |sin(\theta)|^2\). From the trigonometric identity \(cos(x)^2 + sin(x)^2 = 1\) follows that \(|cos(\theta)|^2 + |sin(\theta)|^2 = 1\). For any value \theta, a probability amplitude of \(e^{i \theta}\) results in a probability factor of 1 upon measurement.
Aside from the magnitude of our measurement, which directly tells us which probability we would observe upon measurement, we therefore need another property to uniquely describe the probability amplitude of a quantum state. This property is called phase and is usually expressed as the angle \(\theta \epsilon [0, 2 \pi]\) in the context of the representation as \(e^{i \theta}\).
The role of phase in interference
While phase does not show up in measurement, it influences the evolution of the system up until that point by enabling quantum interference. Even though two states may have the same associated probability, based on their phase they can either amplify or dampen each other.
Let’s take the |+> and |-> states as an example.
\(|+> = \frac{|0> + |1>}{\sqrt{2}}\) \(|-> = \frac{|0> + e^{\pi i} |1>}{\sqrt{2}} = \frac{|0> – 1 |1>}{\sqrt{2}}\)If we measure now, both states result in the same probability distribution. Upon measurement, we would encounter both |0> and |1> in 50% of all cases. Although the magnitudes of each component state of these superpositions are the same, the phases of the |1>-states differ, causing destructive interference.
If we apply a Hadamard gate to an equal superposition state, the following interference pattern illustrates this concept quite neatly.
\(H \frac{1}{\sqrt{2}} (|0> + |1>)\) \(= \frac{1}{\sqrt{2}} (|+> + |->)\) \(= \frac{1}{2} (|0> + |1> + |0> – |1>)\) \(= \frac{1}{2} (2 \times |0>)\) \(= |0>\)For each component state of the superposition, the corresponding |0>- and |1>-states interfere. While the |0>-states amplify each other (constructive interference), the |1>-states (destructive interference) cancel each other out.
All due to a difference in phase.
Applications of phase and interference
While the previous example may seem contrived, purposefully introducing phase differences to cause interference lies at the heart of many quantum algorithms. Below are some high-level examples of quantum algorithms that use phase and interference.
In the Quantum Fourier Transform, phase differences facilitate the transition from the signal to the frequency domain. Depending on the position of a sample in the signal, different phase factors are applied through rotational gates.
Similarly, in Grover’s algorithm, a desired state within a superposition is marked through a relative phase factor of e^{\pi i} = -1. The target state shows the same magnitude as it did before the operation, but the difference in its phase value changes the interference pattern in upcoming operations.
The Phase Estimation algorithm, in contrast, uses Phase Kickback to extract the eigenvalue of any given unitary operator (aka. gate). Since unitary operators are bound to have a determinant whose absolute value is 1 |det(U)| = 1, similar considerations to the one above apply. Just like multiple probability amplitudes can result in the same probability upon measurement, multiple complex eigenvalues can each ensure unitarity. Phase Estimation is a technique for extracting these eigenvalues.
Summary
Phase is a property of quantum systems without a pendant in classical computers. It arises from the distinction between probabilities and probability amplitudes and guides interference. Multiple quantum algorithms use phase to guide interference, including the Quantum Fourier Transform and Grover’s Algorithm.