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arxiv: 2604.15920 · v1 · submitted 2026-04-17 · 🪐 quant-ph · math-ph· math.MP

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Local qubit invariants on quantum computer

Szil\'ard Szalay , Fr\'ed\'eric Holweck
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Pith reviewed 2026-05-10 08:16 UTC · model grok-4.3

classification 🪐 quant-ph math-phmath.MP
keywords quantum circuitslocal unitary invariantsentanglement measuresthree-qubit systemsdirect measurementquantum computingIBM Quantum
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The pith

Two methods enable direct measurement of local unitary invariants on quantum computers.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops two general approaches for constructing quantum circuits that measure local unitary invariants directly rather than through full state reconstruction. Local unitary invariants are quantities unchanged under independent operations on each qubit and include key entanglement measures for three-qubit systems. Implementing these circuits allows assessment of such properties on actual quantum hardware, as shown through demonstrations on the IBM Quantum Platform. A reader would care because direct measurement reduces the need for resource-intensive tomography and supports practical characterization of multipartite entanglement. The work focuses on three-qubit cases but frames the methods as broadly applicable.

Core claim

We present two general methods to implement quantum circuits for the direct measuring of local unitary invariants on quantum computers. We work these out for important three-qubit invariants, and also demonstrate these on the IBM Quantum Platform for important entanglement measures of three qubits.

What carries the argument

Quantum circuits that directly extract local unitary invariants by measuring specific observables tied to the invariant expressions.

Load-bearing premise

The constructed circuits faithfully implement the mathematical definition of the invariants on noisy quantum hardware.

What would settle it

Run the circuits on known three-qubit states such as the GHZ state or W state and compare measured invariant values against exact analytic results; significant deviation would falsify correct implementation.

Figures

Figures reproduced from arXiv: 2604.15920 by Fr\'ed\'eric Holweck, Szil\'ard Szalay.

Figure 1
Figure 1. Figure 1: Quantum circuits implementing the two-qubit norm￾squared (3a). two copies are needed. Index contraction can be done by acting with the bra P1 ij=0 δij ⟨ij|  = ⟨00| + ⟨11| = √ 2 ⟨Bell0| on the respective qubits, which can be achieved by the B† adjoint of the Bell state preparing unitary (9) B = |Bell0⟩ ⟨00| + |Bell1⟩ ⟨01| + |Bell3⟩ ⟨10| + i |Bell2⟩ ⟨11| = C(H ⊗ I), then selecting the 00 outcome of the meas… view at source ↗
Figure 2
Figure 2. Figure 2: Quantum circuits implementing the two-qubit concurrence (3b). and using the outcome 1111 instead of 0000, we end up with (10b) p1111 = | ⟨1111|(B † ⊗ B † )(I ⊗ S ⊗ I)(Uψ ⊗ Uψ)|0000⟩ |2 = c 2 (ψ)/2 2 . Realizing that B† = (H ⊗ I)C, we can get rid of the swap too, as (10c) p1111 = | ⟨1111|(H ⊗ H ⊗ I ⊗ I)(C1,3 ⊗ C2,4)(Uψ ⊗ Uψ)|0000⟩ |2 = c 2 (ψ)/2 2 . (The quantum circuits implementing this method are shown i… view at source ↗
Figure 3
Figure 3. Figure 3: Quantum circuits implementing the three-qubit norm￾squared (14a). permutation, or with CNOTs, and gives p000000 = | ⟨000000| B †⊗3 (S(2,3,5,4) ⊗ I1,6)(Uψ ⊗ Uψ)|000000⟩ |2 = | ⟨000000|(H ⊗3 ⊗ I ⊗3 )(C1,4 ⊗ C2,5 ⊗ C3,6)(Uψ ⊗ Uψ)|000000⟩ |2 = n 4 (ψ)/2 3 . (17) (In this section we draw only the permutation variant of the larger circuits, since it is much more expressive, and the CNOT variant would take up too… view at source ↗
Figure 4
Figure 4. Figure 4: Quantum circuits implementing the three-tangle (14f). where Sσ is the unitary permutation operator implementing the permutation σ = (1, 5, 2)(3)(4, 7, 6)(8, 9, 11, 10)(12), and the CNOTs are Cs = C1,7 ⊗ C2,5 ⊗ C3,6 ⊗ C4,10 ⊗ C8,11 ⊗ C9,12. The invariant g 2 a (ψ) in (14d) can also be implemented in two ways. The index contraction scheme can be read off from (13b) and (14d), in the tensor |γa(ψ)⟩ there are … view at source ↗
Figure 5
Figure 5. Figure 5: Quantum circuits implementing the three-qubit g1(ψ) (14d). B† with the appropriate permutation, or with CNOTs, and gives p111100001111 = | ⟨111100001111|(B †⊗6 )(Sσ)(U ⊗2 ψ ⊗ Uψ ⊗2 )|00 . . . 0⟩ |2 = g1(ψ) 4 /2 6 , (21a) p011011011011 = | ⟨011011011011|(H ⊗4 ⊗ I ⊗3 ⊗ H ⊗2 ⊗ I ⊗3 ) (Cs)(U ⊗2 ψ ⊗ Uψ ⊗2 )|00 . . . 0⟩ |2 = g1(ψ) 4 /2 6 . (21b) Note that in the two variants of the circuit different outcomes giv… view at source ↗
Figure 6
Figure 6. Figure 6: Quantum circuits implementing the three-qubit local con￾currence c1(ψ) (14c). Uψ, U T ψ and U † ψ , a swap, and gives p000000 = | ⟨000000|(U † ψ ⊗ U T ψ )(S(1,4) ⊗ I2,3,5,6)(Y ⊗ I ⊗ I) ⊗2 (Uψ ⊗ Uψ)|000000⟩ |2 = c 4 1 (ψ)/2 2 . (22) Note that (S(2,5) ⊗I1,3,4,6)(I ⊗Y ⊗I) ⊗2 or (S(3,6) ⊗I1,2,4,5)(I ⊗I ⊗Y ) ⊗2 are to be used for c 2 2 (ψ) and c 2 3 (ψ), respectively, because of the permutation covariance of c … view at source ↗
Figure 7
Figure 7. Figure 7: Quantum circuits implementing the three-qubit ω(ψ) (14e). p100100100100 = | ⟨100100100100|(H ⊗4 ⊗ I ⊗3 ⊗ H ⊗2 ⊗ I ⊗3 ) (Cs)(Uψ ⊗ Uψ ⊗ Uψ ⊗ Uψ)|00 . . . 0⟩ |2 = c1(ψ) 4 /2 2+6 . (23b) The invariant ω 2 (ψ) in (14e) can also be implemented in two ways. The index contraction scheme can be read off from (13e) and (14e), in the tensor |T(ψ)⟩ there are contractions with ϵ, then in the norm ∥T(ψ)∥ 2 = δii′δjj′δkk… view at source ↗
Figure 8
Figure 8. Figure 8: Quantum circuits implementing the preparations of the one-parameter families of three-qubit states (26). The usual representing elements of the classes are also covered, |ψ1|23⟩ = |ψ1|23(θ1|23)⟩ = [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Values of the invariants c 2 a (blue), ω 2 (green) and τ 2 (red) for the three one-parameter states (26). The exact values of the invari￾ants ( [PITH_FULL_IMAGE:figures/full_fig_p018_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Measured values of the invariant τ 2 against the calculated values, using the smaller (18) and the larger (19) circuits. and 12 qubits, respectively. The measured values against the calculated values of τ 2 are shown in Figure (10). Here we can see the better performance of the smaller circuit. 4. Summary and remarks We have shown and illustrated two general methods for the implementation of the direct me… view at source ↗
read the original abstract

We present two general methods to implement quantum circuits for the direct measuring of local unitary invariants on quantum computers. We work these out for important three-qubit invariants, and also demonstrate these on the IBM Quantum Platform for important entanglement measures of three qubits.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript presents two general methods for implementing quantum circuits that directly measure local unitary invariants on quantum computers. It derives explicit circuit constructions for key three-qubit invariants and reports an experimental demonstration on the IBM Quantum Platform for three-qubit entanglement measures.

Significance. If the circuit constructions are correct, this work provides practical, hardware-accessible tools for extracting local unitary invariants that characterize entanglement and other properties of quantum states. The explicit three-qubit examples combined with an IBM demonstration constitute a concrete strength, offering a route to avoid full state tomography on near-term devices and enabling direct verification of invariants in experimental settings.

minor comments (2)
  1. The abstract refers to 'important three-qubit invariants' without naming them; adding the specific invariants (e.g., in the first paragraph of the introduction) would improve immediate readability.
  2. Figure captions for the circuit diagrams should explicitly state the gate decomposition and the measurement basis used to extract each invariant.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the recognition that the circuit constructions (if correct) offer practical tools for extracting local unitary invariants without full tomography, and for recommending minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is constructive and self-contained

full rationale

The paper presents two general methods for constructing quantum circuits that directly measure local unitary invariants. These methods are worked out explicitly for three-qubit cases using standard quantum information techniques, followed by an IBM Quantum hardware demonstration for entanglement measures. No load-bearing step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the central claims consist of explicit circuit constructions whose validity is independently testable via the reported experiments. The derivation relies on established quantum circuit primitives rather than renaming known results or smuggling ansatzes via prior self-citations. This is the expected outcome for a methods-and-demonstration paper whose core contribution is the circuit design itself.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are identifiable from the abstract alone.

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Reference graph

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