Quantum Phase Estimation Distribution

Worked Example: Circular Phase Error

Let's calculate the outcome distribution for an eigenphase of 1/3 measured with a 3-bit register (giving 8 possible outcomes).

1. The input state has eigenvalue exp(2πi(1/3)). The register size is N=8.
2. We evaluate the probability of measuring y=3 using Pr(y)=|sum_(k=0)^(N-1)exp(2pi*i*k*(theta-y/N))|²/N².
3. For y=3, the decoded phase is 3/8 = 0.375.
4. The circular phase error is the absolute difference |1/3 - 0.375| = |-0.041666...| = 0.041666...
5. The probability computation gives exactly 0.6878376625896216.

Experiment: Resolving Eigenphases

Follow these steps to observe how phase estimation resolves an eigenphase into a distribution:

1. Open the module at https://riano.app/module/quantum-phase-estimation-distribution.
2. Set the Eigenphase to 1/4 and Register bits to 3.
3. Notice that the peak probability becomes exactly 1.0 at outcome 2, because 2/8 = 1/4 exactly matches the eigenphase.
4. Change the Eigenphase to 1/3. Since 1/3 cannot be represented exactly with 3 bits, the probability distribution spreads, with a peak probability of 0.687... at outcome 3.
5. Increase the Register bits to 4. The peak probability outcome shifts to 5 (decoded as 5/16 = 0.3125), and the circular error decreases.

Limitations and Assumptions

This module computes geometric amplitudes and inverse-QFT probabilities directly; it is not a general quantum computer simulator or sampled success claim. The off-grid phase leaks across all outcomes, and we preserve algebraic zeros rather than tiny trig residues. Most-likely outcome ties are decided by circular distance, tying at the smallest numeric y (e.g., theta 7/8 with N=4 ties y=3 and y=0, and chooses 0). Probe outcome clamps to N-1.

References

Primary references for the mathematical model of quantum phase estimation:

• Quantum Phase Estimation - Berkeley Live Notes: Supports the probability distribution Pr(y) derivation over N states.