Quantum Punctured Phase Estimation

• Quantum Punctured Phase Estimation (QPPE) is a variant of quantum phase estimation that removes qubits corresponding to known eigenphase bits to simplify circuits.
• It uses a punctured inverse QFT and modified control gates to lower qubit and gate counts, enhancing resource efficiency in hybrid HHL algorithms.
• This method leverages preprocessed eigenvalue information, enabling significant error reduction on NISQ devices without compromising estimation precision.

Searching arXiv for the cited QPPE paper and closely related phase-estimation variants to ground the article in current literature. Quantum punctured phase estimation (QPPE) is a variation of quantum phase estimation in which qubits corresponding to already known eigenphase bits are removed from the phase register, and the gates that would have acted on or been controlled by those qubits are correspondingly omitted or simplified. In the formulation introduced in "Two Variations of Quantum Phase Estimation for Reducing Circuit Error Rates: Application to the Harrow--Hassidim--Lloyd Algorithm" (Lee et al., 9 Jul 2025), QPPE is a circuit-structural method for reducing qubit count, gate count, and overall circuit error rates, especially inside hybrid implementations of the Harrow--Hassidim--Lloyd (HHL) algorithm. The term also has broader, looser conceptual relatives in the literature, where “punctured” can refer to selective, windowed, or domain-restricted phase estimation rather than bit-level removal of known phase-register qubits (Moore et al., 2021).

1. Definition and position within the phase-estimation family

Standard quantum phase estimation considers a unitary $U$ with eigenstate $\ket{u}$ and eigenphase $\varphi\in[0,1)$ such that

$U\ket{u} = e^{2\pi i \varphi}\ket{u}.$

Writing

$$\varphi = 0.\varphi_1 \varphi_2 \dots \varphi_n \dots \qquad (\varphi_j\in\{0,1\}),$$

standard $n$-qubit QPE uses a phase register $F$ of $n$ qubits and a system register $G$ holding $\ket{u}$. After Hadamards and controlled powers $\ket{u}$0, the phase register carries the usual phase-encoded product state, and inverse QFT without SWAPs, denoted IQFT$\ket{u}$1, maps this to the measured bit string for the phase estimate (Lee et al., 9 Jul 2025).

QPPE assumes that some phase bits are already known. If

$\ket{u}$2

is the set of known bit positions, and $\ket{u}$3 is known for each $\ket{u}$4, then the central observation is that the $\ket{u}$5-th qubit of the phase register is used precisely to estimate bit $\ket{u}$6. If $\ket{u}$7 is known a priori, the corresponding phase-register qubit and the gates that encode or decode that bit can be removed or simplified while still correctly estimating the remaining unknown bits. In this sense, puncturing means removing the wire for qubit $\ket{u}$8, removing gates targeting it, and replacing gates that would have used it as a control by either omission or unconditional single-qubit rotations depending on whether the known bit is $\ket{u}$9 or $\varphi\in[0,1)$0 (Lee et al., 9 Jul 2025).

A common misconception is that QPPE is merely truncation to fewer least- or most-significant bits. The construction is more specific: it permits a non-contiguous subset of unknown phase bits, with the known positions handled classically and reinserted after measurement. That distinguishes QPPE from simply lowering precision or stopping an iterative scheme early (Lee et al., 9 Jul 2025).

2. Circuit transformation and the punctured inverse QFT

The circuit-level implementation takes as input a black box for controlled-$\varphi\in[0,1)$1, an eigenstate $\varphi\in[0,1)$2, and an index set $\varphi\in[0,1)$3 of known bits together with their values. In the modified phase-encoding stage, for each $\varphi\in[0,1)$4, one applies a Hadamard and controlled-$\varphi\in[0,1)$5 only if $\varphi\in[0,1)$6; if $\varphi\in[0,1)$7, one does nothing to that qubit, and in practice the qubit need not be allocated at all (Lee et al., 9 Jul 2025).

The inverse transform is a punctured IQFT$\varphi\in[0,1)$8. It is implemented recursively from the bottom up. For the last qubit $\varphi\in[0,1)$9, if $U\ket{u} = e^{2\pi i \varphi}\ket{u}.$0, one applies $U\ket{u} = e^{2\pi i \varphi}\ket{u}.$1. For each $U\ket{u} = e^{2\pi i \varphi}\ket{u}.$2, if $U\ket{u} = e^{2\pi i \varphi}\ket{u}.$3, then for every $U\ket{u} = e^{2\pi i \varphi}\ket{u}.$4, a known bit at position $U\ket{u} = e^{2\pi i \varphi}\ket{u}.$5 is treated classically: if $U\ket{u} = e^{2\pi i \varphi}\ket{u}.$6 and $U\ket{u} = e^{2\pi i \varphi}\ket{u}.$7, a single-qubit $U\ket{u} = e^{2\pi i \varphi}\ket{u}.$8 is applied on qubit $U\ket{u} = e^{2\pi i \varphi}\ket{u}.$9; if $$\varphi = 0.\varphi_1 \varphi_2 \dots \varphi_n \dots \qquad (\varphi_j\in\{0,1\}),$$0 and $$\varphi = 0.\varphi_1 \varphi_2 \dots \varphi_n \dots \qquad (\varphi_j\in\{0,1\}),$$1, no gate is needed; if $$\varphi = 0.\varphi_1 \varphi_2 \dots \varphi_n \dots \qquad (\varphi_j\in\{0,1\}),$$2, the usual controlled-$$\varphi = 0.\varphi_1 \varphi_2 \dots \varphi_n \dots \qquad (\varphi_j\in\{0,1\}),$$3 is applied. Finally, $$\varphi = 0.\varphi_1 \varphi_2 \dots \varphi_n \dots \qquad (\varphi_j\in\{0,1\}),$$4 is applied to qubit $$\varphi = 0.\varphi_1 \varphi_2 \dots \varphi_n \dots \qquad (\varphi_j\in\{0,1\}),$$5. If $$\varphi = 0.\varphi_1 \varphi_2 \dots \varphi_n \dots \qquad (\varphi_j\in\{0,1\}),$$6, nothing is done (Lee et al., 9 Jul 2025).

The operational rule is therefore simple. A known control bit equal to $$\varphi = 0.\varphi_1 \varphi_2 \dots \varphi_n \dots \qquad (\varphi_j\in\{0,1\}),$$7 deletes a controlled gate because the control never fires. A known control bit equal to $$\varphi = 0.\varphi_1 \varphi_2 \dots \varphi_n \dots \qquad (\varphi_j\in\{0,1\}),$$8 converts that controlled gate into an unconditional single-qubit phase gate on the target. Only the unknown-bit qubits are measured, and the full $$\varphi = 0.\varphi_1 \varphi_2 \dots \varphi_n \dots \qquad (\varphi_j\in\{0,1\}),$$9-bit phase string is reconstructed classically by reinserting the known bits at positions $n$0 (Lee et al., 9 Jul 2025).

This circuit transformation is the defining mechanism of QPPE. It reduces the active phase register from $n$1 qubits to $n$2 qubits, and it replaces some entangling structure in IQFT$n$3 by single-qubit phases or complete omission. The reduction is structurally targeted rather than uniform across the register (Lee et al., 9 Jul 2025).

3. Mathematical structure, correctness, and relation to shifted phase estimation

The correctness argument parallels standard QPE. Let

$n$4

with known-bit set $n$5 and unknown-bit set $n$6. In the phase-encoding stage, controlled-$n$7 is applied only for $n$8. The known bits contribute a classically computable phase factor rather than a quantum-controlled one. Because the inverse QFT is linear, if a control qubit is always in a known state $n$9, then a controlled rotation is equivalent to either the identity or an unconditional rotation. Replacing it accordingly preserves the intended mapping from the phase-encoded amplitudes to the computational basis. The paper states that the distribution over the unknown bits has the same form as in standard QPE, now with output length $F$0 (Lee et al., 9 Jul 2025).

For standard QPE, to get $F$1 correct bits with failure probability $F$2, one needs

$F$3

qubits in the phase register. For QPPE, if only the $F$4 unknown bits are to be estimated, the same precision relation is used with the effective bit length: $F$5 Conceptually, standard QPE, quantum shifted phase estimation (QSPE), and QPPE share the same precision theory; they differ in which subset of bits is estimated, or how the estimate is shifted (Lee et al., 9 Jul 2025).

QPPE was introduced together with QSPE. QSPE estimates a contiguous block of bits beginning at position $F$6 by left-shifting the phase via powers $F$7, so that the fractional part of $F$8 reveals $F$9. QPPE differs in that it assumes prior knowledge of an arbitrary subset of bits, not necessarily contiguous, and removes those positions from the circuit altogether. The two techniques are independent and can be combined; in the hybrid HHL construction they act on disjoint subsets of qubits, so their order does not matter (Lee et al., 9 Jul 2025).

At the level of black-box complexity, puncturing does not evade the general lower-bound structure of phase estimation. A separate query-complexity analysis proves that for any $n$0 and $n$1, every algorithm requires at least $n$2 queries to obtain an $n$3-approximation for the phase with probability at least $n$4 (Lin, 2023). This suggests that QPPE is best understood as a resource-reallocation and circuit-simplification technique rather than a violation of the usual information-theoretic scaling.

4. QPPE inside the hybrid HHL algorithm

In HHL, one solves

$n$5

with $n$6 Hermitian and pre-scaled so its eigenvalues lie in $n$7. The unitary used is

$n$8

If

$n$9

then the phase-estimation step maps

$G$0

followed by a controlled rotation on an ancilla qubit and then inverse phase estimation to uncompute $G$1 (Lee et al., 9 Jul 2025).

The hybrid method uses a separate preprocessing loop based on repeated QPE runs to learn eigenvalue patterns. Each run produces one row of an $G$2 binary matrix

$G$3

whose rows correspond to estimated eigenvalue bits. Classical processing then finds a distinguishing column set $G$4 such that the restricted rows are all distinct. A minimal such set is called a minimal distinguishing set. The smallest-index qubit in $G$5 is the leading qubit. Columns outside $G$6 are classified as constant or non-constant, and their positions relative to the leading qubit determine whether QSPE or QPPE can remove them (Lee et al., 9 Jul 2025).

The modified HHL subroutines are denoted PE$G$7, CR$G$8, and IPE$G$9. In PE$\ket{u}$0, qubits before the leading one are removed via QSPE because they are non-distinguishing, and constant qubits after the leading one are punctured via QPPE. Non-constant, non-distinguishing qubits after the leading one are retained in PE$\ket{u}$1 and IPE$\ket{u}$2 because they still affect the phase evolution, but they are not needed as controls in CR$\ket{u}$3. The controlled-rotation stage uses only the distinguishing qubits $\ket{u}$4 as controls. In this workflow, QPPE is precisely the component that punctures away constant known bits, simplifying both PE/IPE and the control structure of the rotation stage (Lee et al., 9 Jul 2025).

The resulting qubit taxonomy is central to the method. Distinguishing qubits are used in PE, CR, and IPE. Constant qubits after the leading one can be omitted in all steps via QPPE. Non-constant qubits after the leading one remain relevant to PE and IPE but not to CR. Non-distinguishing qubits before the leading one can be omitted everywhere via QSPE (Lee et al., 9 Jul 2025).

5. Resource reductions and experimental demonstrations

The principal resource claim is that puncturing known bits reduces both the active phase-register size and the entangling-gate load. In the worked hybrid-HHL example for general $\ket{u}$5, the original HHL and the earlier Hybrid19 method use $\ket{u}$6 qubits in the phase register during PE and IPE, whereas Hybrid25, which uses QSPE and QPPE, uses 3 qubits in the phase register. The same example reduces Hadamards in PE+IPE from $\ket{u}$7 to 12, controlled-$\ket{u}$8 from $\ket{u}$9 to $\ket{u}$00, and controlled-$\ket{u}$01 from $\ket{u}$02 to 6 (Lee et al., 9 Jul 2025).

On IBM hardware, the paper compares Hybrid19 and Hybrid25 for the same small linear system. Hybrid19 uses a 6-qubit phase register and 9 total qubits; Hybrid25 uses a 3-qubit phase register and 6 total qubits. The transpiled circuits on ibm_kingston show the following counts (Lee et al., 9 Jul 2025):

Method	Total qubits	Transpiled gates
Hybrid19	9	650 single-qubit, 189 cz
Hybrid25	6	193 single-qubit, 54 cz

The detailed counts are $\ket{u}$03 sx, $\ket{u}$04 rz, and $\ket{u}$05 x for Hybrid19, versus $\ket{u}$06 sx, $\ket{u}$07 rz, and $\ket{u}$08 x for Hybrid25. The paper summarizes this as roughly $\ket{u}$09 fewer single-qubit sx gates, roughly $\ket{u}$10 fewer rz gates, and roughly $\ket{u}$11 fewer two-qubit cz gates. These reductions are attributed to eliminating constant qubits via QPPE in PE/IPE, limiting CR$\ket{u}$12 to the distinguishing qubits, and skipping non-distinguishing leading bits via QSPE (Lee et al., 9 Jul 2025).

The target normalized solution vector for the test system is reported as

$\ket{u}$13

with ideal probabilities on register $\ket{u}$14 approximately

$\ket{u}$15

AerSimulator runs show that Hybrid19 and Hybrid25 agree with the ideal HHL circuit in the noiseless setting. On ibm_kingston, using 4096 shots and postselection on the ancilla outcome $\ket{u}$16, Hybrid25 produces a histogram visibly closer to the theoretical distribution than Hybrid19. Extra unwanted outcomes remain, but with smaller probability in Hybrid25. In the paper’s interpretation, the qubit and gate reductions therefore translate into practical error mitigation on current superconducting hardware (Lee et al., 9 Jul 2025).

6. Assumptions, limitations, and related meanings of “punctured” estimation

QPPE depends on reliable prior information about some phase bits. In the hybrid HHL construction, that information is obtained from a separate preprocessing loop that repeatedly runs QPE, constructs the binary matrix $\ket{u}$17, and classifies columns as distinguishing, constant, or non-distinguishing. If the prior classification is wrong, then the circuit hard-codes an incorrect bit value and systematic error results. The paper also notes that finding a minimal distinguishing column set is NP-hard in general, that such a set is not unique, and that no formal complexity proof is given that the hybrid method preserves the exponential speedup of HHL in the fully general setting (Lee et al., 9 Jul 2025).

The regime of usefulness is correspondingly specific. The method is most attractive when the number of distinct relevant eigenvalues is much smaller than $\ket{u}$18, so that many phase bits are constant or redundant for distinguishing eigenvalues, when hardware is NISQ-like and two-qubit gate errors are dominant, and when one can afford offline spectral preprocessing for a fixed matrix $\ket{u}$19 and then reuse the streamlined circuit many times (Lee et al., 9 Jul 2025).

The phrase “quantum punctured phase estimation” also has broader conceptual relatives in the literature, but these should be distinguished from the bit-level construction above. In "Statistical Approach to Quantum Phase Estimation" (Moore et al., 2021), the term itself is not used, but the statistical phase estimation algorithm naturally supports restricted, selective, or windowed phase estimation by limiting the trial-phase range $\ket{u}$20 and excluding known state-space directions. There, “puncturing” refers to focusing on chosen regions of the phase spectrum or chosen eigen-directions rather than removing known phase-register bits. A related randomized statistical phase-estimation scheme has also been described as using a random, sparse, “punctured view” of Hamiltonian dynamics, because each circuit samples only a small number of Pauli rotations while global spectral information is reconstructed statistically from many shallow runs (Wan et al., 2021). In optical metrology with squeezed vacuum, phase domains can be effectively restricted to $\ket{u}$21 or $\ket{u}$22 by state and measurement symmetries; this is another distinct sense in which phase estimation can be “punctured,” now at the level of identifiable phase intervals rather than qubit removal (Rodríguez-García et al., 2023).

Taken together, these usages indicate two distinct meanings. In the strict circuit-theoretic sense introduced in (Lee et al., 9 Jul 2025), QPPE is a non-contiguous, bit-selective modification of QPE that hard-codes known bits and removes their qubits. In a looser conceptual sense, the literature also uses “punctured” to denote selective access to only part of the phase domain, part of the spectrum, or part of the Hamiltonian dynamics. The former is a concrete algorithmic variant of QPE; the latter is a broader design philosophy for partial or localized phase inference.