phase estimation

In plain terms

A quantum computation's answer often arrives as a phase. Every amplitude in a quantum state is a number with a size and an angle, the angle is its phase, and no direct look at the state reveals it, because probabilities depend only on sizes, so the task is to read an angle you can't see, using a register with finitely many dial positions.

Phase estimation is the reading procedure. Apply the operation whose phase you want repeatedly, with the number of repetitions controlled by the position of the register, so the unknown angle winds a different number of times into each position, and then run the quantum Fourier transform backwards. The interference between the windings makes the amplitude concentrate at the dial position closest to the true angle, and if the angle sits exactly on a tick you read it with certainty.

Usually the angle sits between ticks, and the textbook promise is that the nearest tick still comes up with probability at least four divided by pi squared, a little over forty percent, whatever the size of the register. That constant comes from the Dirichlet kernel, the standard formula for how a sum of evenly spaced oscillations concentrates, and a fixed floor is what makes the subroutine composable, as whatever consumes the reading can budget for a guaranteed rate of success.

In CSD

Both halves of the textbook result are theorems in the corpus, in the CSD-free library tree where an outside project can import them, the exact case, in which an angle that sits on a tick is read with certainty by inverting the Fourier transform, and the fixed floor of four divided by pi squared for an arbitrary angle.

The result was first proved inside the corpus's development of Shor's algorithm and later extracted into a module of its own, once it was noticed that its statements never mention orders or modular arithmetic. The module is generic in the register size, and Shor's order-finding consumes it. The standing disclaimer for the algorithm layer applies, as these results rest on the programme's unitary-evolution pillar and don't support it or the foundations.

Mathematically

On the T-level counting register, the phase state (1/sqrt T) sum over x of e^{2 pi i phi x} |x> read by the inverse QFT at index c has amplitude the Dirichlet sum (1/T) sum over x of e^{2 pi i (phi - c/T) x}, the geometric series closes it, and the norm reduces to T^{-2} sin^2(pi delta T) / sin^2(pi delta) with delta = phi - c/T. On resonance the probability is one (phase_estimation_exact, an inverse-QFT inversion).

For |delta| <= 1/(2T), Jordan's inequality bounds the numerator below and |sin t| <= |t| the denominator above, giving probability at least 4/pi^2 at the closest index (phase_estimation_lower_bound, Nielsen and Chuang section 5.2). A two-index refinement is proved as well, as the two grid points straddling the true angle carry at least 8/pi^2 between them (dirichlet_two_index, phase_estimation_two_index), which is the constant BHMT's amplitude-estimation theorem needs. The module lives in the Cat-1 tree, extracted in 2026 from the Shor development, and Shor's order- finding consumes it at phase s/r. It's a single-phase statement only, so a consumer racing several eigenvalue branches must control its own cross-terms.

Related entries

• quantum Fourier transformThe quantum Fourier transform runs Fourier's transform as a single reversible motion of a quantum register, and it's the lens that turns hidden periods and phases into outcomes a measurement can land on.
• amplitude estimationPoint phase estimation at the rotation inside amplitude amplification and the success probability itself becomes measurable, quadratically faster than sampling could ever learn it.
• Shor's algorithmShor's algorithm factors large numbers in polynomial time on a quantum computer, and it turned quantum computing from a curiosity into a budget line and made every deployed cryptosystem provisional.
• Fisher informationThere's a floor under every error bar, classical or quantum, and Fisher information is what sets it.
• Hadamard testThe Hadamard test turns a matrix element, a complex number buried inside an operator, into the bias of a single coin.