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Permutation-Invariant N-body gates via Tavis-Cummings Hamiltonian

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper proves that any quantum gate that treats all qubits alike—including the multi-qubit Toffoli-class controlled-Z—can be built from the Tavis-Cummings interaction plus global x and z fields, with no individual qubit addressing.

desk verdict Strong global-control result for permutation-invariant gates, with one load-bearing external lemma that needs referee scrutiny. read the letter →

arxiv 2506.03453 v3 pith:W7MCWXLW submitted 2025-06-03 quant-ph math-phmath.MPnucl-thphysics.atom-phphysics.optics

Plato Deliyannis , Iman Marvian This is my paper
classification quant-phmath-phmath.MPnucl-thphysics.atom-phphysics.optics PACS 03.67.Lx42.50.Pq
keywords Tavis-CummingsHamiltonianpermutation-invariantgatesglobalquantumcontrolmulti-qubitToffolisymmetricsubspacestatepreparationDickestatesaccidentalsymmetrytwo-oscillatorangular-momentummodel
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper aims to prove that global control, with no individual qubit addressing, is enough to run every unitary operation that treats all qubits alike. Its main theorem says that the Tavis-Cummings Hamiltonian—$n$ qubits identically coupled to a single bosonic mode—together with uniform global $x$ and $z$ fields, realizes every permutation-invariant $n$-qubit unitary up to a global phase, with the oscillator beginning and ending in vacuum. Because the $n$-qubit controlled-$Z$ gate is permutation-invariant and is equivalent up to Hadamards to the multi-qubit Toffoli gate, the result gives a global-control route to a key multi-qubit primitive. A corollary is that every state in the symmetric subspace, including GHZ and Dicke states, can be prepared with the same resources. The paper also fully characterizes the smaller set of operations reachable with only a global $z$ field, uncovering an accidental symmetry of the TC interaction, and gives explicit two-qubit pulse sequences for CZ, SWAP, iSWAP, and $\sqrt{i\text{SWAP}}$.

What carries the argument

The load-bearing object is the Tavis-Cummings Hamiltonian $H_{\mathrm{TC}}=g_{\mathrm{TC}}(J_+ a+J_- a^\dagger)$, the collective version of the Jaynes-Cummings interaction in which all qubits are identically coupled to a single oscillator. It conserves the charge $Q=J_z+a^\dagger a+n/2$, splitting the infinite-dimensional Hilbert space into finite-dimensional sectors $H_{q,j}$ labelled by charge and total angular momentum; this block structure makes exact characterization possible. Inside each sector, $H_{\mathrm{TC}}$ together with its phase-shifted partner generates the full special unitary group, so the only obstacles to universality come from correlations between sectors. The paper shows the only cross-sector correlations are relative-phase constraints and an accidental symmetry pairing sectors $(q,j)$ and $(q',j')$ with $q=n/2+2j'-j$ and $q'=n/2-j'+2j$, understood through the two-oscillator model of angular momentum, where $J_+\otimes a$ becomes $b_1^\dagger b_2 a$ and is invariant under swapping the physical oscillator with the second virtual oscillator. A global $x$ field breaks the charge conservation and upgrades the z-only classification into the full permutation-invariant universality of Theorem 1.

What would settle it

For $n=3$ qubits, simulate or measure $e^{-iH_{\mathrm{TC}}t}$ and compare the two-by-two blocks acting on $H_{q=1,j=3/2}$ and $H_{q=4,j=1/2}$: the paper predicts these blocks are exactly equal for all $t$, so any discrepancy invalidates the accidental-symmetry step behind Theorem 8. A second check is to run a numerical optimal-control search for the $n$-qubit controlled-$Z$ gate (any $n\ge4$) under only $H_{\mathrm{TC}}$ and global $x,z$ fields with the oscillator required to return to vacuum; failure to find a sequence at growing resource bounds would contradict the existence claim.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 1: all permutationally-invariant unitary transformations on $n$ qubits can be realized, up to a global phase, by $H(t)=c(t)H_{\mathrm{TC}}+z(t)H_z+x(t)H_x$ with $H_{\mathrm{TC}}=g_{\mathrm{TC}}(J_+ a+J_- a^\dagger)$ and the oscillator in vacuum. The result is not an immediate consequence of the symmetries: an earlier no-go theorem limits what symmetric $k$-body gates can do for $k<n$, and $H_{\mathrm{TC}}$ has an accidental symmetry that further constrains the z-field-only dynamics. The supporting technical result, Theorem 8, fully characterizes the z-field-only group: inside each charge–angular-momentum sector essentially every unitary is reachable, and the only cross-sector constraints are a relative-phase rule and equality of components in sector pairs linked by the accidental symmetry. This characterization explains why $n=2$ and $3$ are special—with a z-field alone all PI U(1)-invariant unitaries are reachable—while for $n\ge4$ some are not, including the multi-controlled CZ gate; adding the global $x$ field restores full universality.

Load-bearing premise

The load-bearing premise is that Theorem 8's characterization of the z-field-only group is complete; that proof imports a semi-universality lemma from a companion paper, and the argument also assumes an ideal single-mode Tavis-Cummings interaction with identical qubit couplings in the rotating-wave approximation.

Editorial extensions

If this is right

  • All permutation-invariant $n$-qubit gates, including the $n$-qubit controlled-$Z$ gate equivalent to the multi-qubit Toffoli gate, can be executed with global fields only, removing the need for individual qubit addressing in architectures with a shared oscillator mode.
  • GHZ states, Dicke states, and any other state in the symmetric subspace can be prepared from $|0\rangle^{\otimes n}$ with the oscillator in vacuum using the same resources.
  • For two qubits, explicit pulse sequences realize CZ, SWAP, iSWAP, and $\sqrt{i\text{SWAP}}$ with interaction times of about $2.87$, $1.27$, $2.55$, and $2.68$ times $2\pi/g_{\mathrm{TC}}$; arbitrary z-conserving permutation-invariant two-qubit gates need at most about $3.92\times 2\pi/g_{\mathrm{TC}}$.
  • With the z-field alone the reachable set is strictly smaller for $n\ge4$: the multi-qubit controlled-$Z$ with three or more controls is not reachable, while the anti-controlled-$Z$ gate is.
  • A universal qubit-oscillator SWAP gate exists within the symmetric subspace, allowing preparation of arbitrary oscillator states with up to $n$ excitations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper: because the construction uses only the algebraic form of $H_{\mathrm{TC}}$, the same circuits should transfer to any platform with one collective mode and identical coupling, such as circuit QED or cavity QED; the paper notes such platforms but does not work out pulse-level noise performance.
  • Beyond the paper: the accidental symmetry gives a ready-made diagnostic—monitoring the predicted equality of paired-sector blocks under $H_{\mathrm{TC}}$ would test whether a device really implements the ideal collective interaction.
  • Beyond the paper: the qubit-oscillator SWAP module points to a hybrid discrete-continuous-variable protocol—encode a state in the symmetric subspace, swap it into the oscillator, then process it there—whose resource overhead the paper does not quantify.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper considers n qubits identically coupled to a single bosonic mode via the Tavis-Cummings interaction, together with global uniform x and z fields, with the oscillator initialized and returned to vacuum. The central theorem (Theorem 1) asserts that every permutation-invariant (PI) unitary on n qubits can be realized by these controls, up to a global phase. A corollary extends the result to preparation of all PI states, including GHZ and Dicke states. The paper also characterizes the group V_z of unitaries realizable with the TC interaction and a global z field (Theorem 8), identifies an accidental symmetry of H_TC that is explained through Schwinger's oscillator model, and provides explicit two-qubit pulse sequences for CZ, SWAP, iSWAP, sqrt(iSWAP), and related gates, with interaction-time tables. The proof of Theorem 1 is built from Theorem 8, a decomposition lemma, and a reduction to global rotations.

Significance. If the central claim is valid, the result is significant: all PI unitaries, including n-qubit controlled-Z gates, are implementable with global control only, using a single oscillator mode in its vacuum state. This addresses a question left open in prior work and provides concrete resource estimates for two-qubit gates. The manuscript has notable strengths: the appendices are detailed, with explicit matrix elements, variance computations, and numerical circuit parameters; the two-qubit constructions were checked by explicit matrix multiplication; and the accidental-symmetry analysis is an original contribution. However, the universality theorem is not self-contained at its core: the sufficiency direction of Theorem 8, and hence the completeness direction of Theorem 1, depends on Lemma 12 quoted from the same-group preprint [50], which is not proved in this manuscript. The significance of the paper is therefore conditional on that external result or on a supplied proof.

major comments (3)
  1. [Section VIII; Appendix G.1; Lemma 12] The sufficiency direction of Theorem 8, and hence the completeness half of Theorem 1, rests on Lemma 12 of [50] (called Theorem 12 in Section VIII), which is quoted in Appendix G.1 but not proved in this manuscript. This lemma is load-bearing: Proposition 13, which rules out all correlations between sectors other than the accidental pairs, is obtained by applying Lemma 12 to the isotypic decomposition, and no independent proof of semisimple-part universality is provided. Because [50] is an arXiv preprint from the same research group and its stated setting (semi-universality of 3-qudit SU(d)-invariant gates) differs from the U(1) x S_n, infinite-dimensional, multiplicity-carrying setting used here, the proof as submitted is not self-contained at a central point. The authors should include a proof of Lemma 12 or of a version specialized to their isotypic decomposition, or supply a complete proof of Proposition 13.
  2. [Appendix G.3; Proposition 13] The proof of Proposition 13 selects, for each accidental pair, one sector and invokes Lemma 12's conditions A and B' on the remaining sectors; two points need explicit verification. First, the reduction from B' to B uses a Taylor expansion of |Tr(pi_lambda(e^{-iHt}))|^2; the argument should state why the trace and derivative are well-defined for the truncated infinite-dimensional setting and why the relevant H can be taken in the Lie algebra generated by H_TC and J_z. Second, condition A in Lemma 12 concerns the multiplicity subsystem M_lambda in the isotypic decomposition Eq. (G1); in the application, the multiplicity spaces C^{m(n,j)} are nontrivial for j < n/2, and the manuscript does not spell out how Q_lambda and M_lambda are assigned for the U(1) x S_n decomposition, nor why the per-sector SU(H_{q,j}) universality proved in Appendix F implies the lemma's SU(M_lambda) condition under that assignment. These are verification gaps, not demonstrated errors, but they are load-bearing for Theorem 8.
  3. [Section III B; Appendix J; Lemma 5] The proof that all PI unitaries are generated by global x-rotations together with U(1)-invariant PI unitaries is given by a Lie-algebra argument that ends with 'it is straightforward to show, by recursively taking commutators, that the operators in Eq. (J10) generate all permutationally-invariant skew-Hermitian operators.' No recursion or alternative argument is supplied. Since this lemma is an explicit step in the proof of Theorem 1, the derivation should be written out, or a precise reference should be given. This issue is more readily fixable than the Lemma 12 dependence, but it is still part of the load-bearing chain.
minor comments (5)
  1. [Section III B and Appendix H] The text refers to 'Theorem 4' for the characterization proven in Appendix H, but the formal statement in Section III A is Lemma 4; the numbering should be harmonized.
  2. [Section VIII and Appendix G] The external result is called 'Theorem 12' in Section VIII and 'Lemma 12' in Appendix G.1; please use one name and clearly mark it as a quoted result.
  3. [Appendix G.5] The text refers to 'Theorem 13' where the statement being used is Proposition 13 from Appendix G.3; this should be corrected.
  4. [Section VI C, Eq. (58)] The definition of the commutator subgroup uses V both for the group and for a generic unitary in the definition; using a different symbol for the group would remove ambiguity.
  5. [Tables II, III and Figures 2, 4] The captions contain editorial remnants such as 'Correct version' and 'Previous figure, with j, j' switched'; these should be cleaned before publication.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction: the universality theorem is derived from the bare TC Hamiltonian; the only same-group citation (Lemma 12 of [50]) is a general external lemma, so it is a verification dependency rather than a circular one.

full rationale

The derivation chain is Theorem 1 -> Lemma 5 + Theorem 4 -> Theorem 8. Theorem 8's necessity is obtained by explicit matrix elements of H_TC and J_z (Appendix G.4, Eq. (G16)); its sufficiency uses subsystem universality (Appendix F, from the external Lemma 10 of [61]) plus the semi-universality criterion Lemma 12 quoted from [50] (Appendix G.1). Lemma 12 is not proved in this paper and is by the same group, but it is a parameter-free group-theoretic statement about arbitrary G-invariant unitaries, not a statement that assumes the TC characterization. Thus no equation is equal to its own input by construction, and no fitted parameter is renamed as a prediction. The fixed F/A gate angles are analytic or numerically checked constants (Appendix K.5, K.7, Table VI), and the two-qubit circuits are verified by explicit matrix multiplication. The only caveat is that the completeness half of Theorem 8 is as strong as Lemma 12 of [50]; if that lemma were false or inapplicable in this infinite-dimensional, multiplicity-carrying setting, Theorem 8 and hence Theorem 1 could fail. That is a correctness/verification risk, not circularity. Score 2 reflects the presence of same-group citations in the proof and motivation, but with independent content and no circular reduction.

Assumptions & free parameters 2 free parameters · 6 assumptions · 1 invented entities

The central theorems are parameter-free derivations from the TC Hamiltonian under standard controllability assumptions; no data are fitted and no target result is assumed. The numerical constants in the paper (F-gate angles, A-gate Euler angles) are exact or numerically solved solutions of synthesis equations and affect only the explicit two-qubit circuits, not the theorems. The heaviest external dependency is Lemma 12 of the same-group preprint [50], on which the completeness of Theorem 8 rests; per-sector subsystem universality was previously established by Keyl et al. [24], and the controllability tool comes from Schirmer, Fu and Solomon [61]. The only introduced mathematical object is the conserved operator S (accidental symmetry), which is derived from the explicit matrix elements of H_TC and independently explained by the Schwinger oscillator picture; it is not a postulated physical entity.

free parameters (2)
  • F-gate angles phi_0, phi_1 (Eq. K58) = phi_1 = (1/2) arccos(7/16); phi_0 = arctan(-sqrt(23)/3) + pi
    Introduced in Section K7 to make the F-gate identity (K56) hold. They are exact solutions of a determined system, not data fits, and they affect only the explicit two-qubit circuits, not the main theorems.
  • A-gate Euler angles (Table VI) = numerical, 8 decimal places
    Angles solving the exact synthesis equations (K46) and (K50) for each target gate; they are construction parameters for the explicit circuits, deterministically solved, not fitted to experimental data.
assumptions (6)
  • domain assumption The qubit-oscillator system is described by the single-mode Tavis-Cummings Hamiltonian with identical coupling g_TC (Eq. 2), in the rotating frame that removes the oscillator frequency.
    Physical platforms realize the TC model only under the rotating-wave approximation with identical coupling and one dominant mode; all theorems are proved for this idealized Hamiltonian.
  • domain assumption The control set consists of arbitrary piecewise-constant switching of H_TC, H_z and H_x, with at most one Hamiltonian active at a time (Eqs. 3-4).
    Standard quantum-control controllability setup; the reduction of control strengths to {0, +/-1} is a known fact cited from [66].
  • domain assumption Lemma 12 of [50] (semi-universality criterion for G-invariant unitaries) is correct and applicable.
    Theorem 8's completeness proof and hence the phase conditions underlying Theorem 1 rest on this lemma from a same-group preprint, quoted in Appendix G but not proved in this paper.
  • standard math The controllability result of Schirmer, Fu and Solomon [61] (used as Theorem 10/11) holds as stated.
    Used in Appendix F to prove subsystem universality (SU(H_q,j)) within each sector; the paper notes Keyl et al. [24] proved this earlier.
  • domain assumption The oscillator is initialized in and returned to the vacuum state, and the x-field is applied only when the oscillator is back in vacuum (Eq. 5).
    This defines what it means to realize a qubit-only gate in this model and is assumed throughout Sections III to VIII.
  • standard math Schur-Weyl duality and the sector decomposition of (C^2)^n into C^{m(n,j)} x C^{2j+1} blocks (Eqs. 46-51).
    Standard representation theory used for the sector decomposition and Schur's lemma arguments.
invented entities (1)
  • Accidental symmetry operator S (Eqs. 70-71) independent evidence
    purpose: A conserved observable of H_TC that pairs sectors (q,j) and (q',j') with identical H_TC matrix elements; it explains the restrictions on the z-only reachable group in Theorem 8.
    S is derived from and directly checkable against the explicit matrix elements of H_TC; the commutation [H_TC, S] = 0 is proven in Appendix E and independently explained via Schwinger oscillators (Eqs. 84-86). It is a mathematical conserved quantity, not a postulated physical field, and its consequences (sector-pair restrictions) are falsifiable in any TC device.

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Pith. "Pith review of Permutation-Invariant N-body gates via Tavis-Cummings Hamiltonian." pith.science (2026). https://pith.science/paper/W7MCWXLW

@misc{pith2026250603453,
  author       = {Pith},
  title        = {Pith review of: Permutation-Invariant N-body gates via Tavis-Cummings Hamiltonian},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W7MCWXLW}},
  note         = {Machine review of arXiv:2506.03453}
}
abstract

Global control provides a promising route to implementing multi-qubit gates without individual qubit addressing. This is especially appealing for permutation-invariant (PI) gates, whose symmetry is often broken when they are compiled into individually addressed one- and two-qubit gates. Important examples include SWAP, $\sqrt{i\text{SWAP}}$, and the n-qubit controlled-Z gate, which is equivalent, up to two single-qubit Hadamard gates, to the multi-qubit Toffoli gate. Motivated by this global-control perspective, we show that all PI unitaries on an arbitrary number of qubits can be realized using the Tavis-Cummings (TC) interaction, the multi-qubit version of the Jaynes-Cummings interaction, together with global uniform z and x fields. Here, the $n$ qubits are identically coupled to a single bosonic mode (oscillator), which is initialized in and returned to its vacuum state. A corollary is that all PI states, including GHZ and Dicke states, can be prepared using the same global control. For the case n=2 qubits, which is particularly important in quantum computing, we also find explicit pulse sequences for implementing all PI qubit unitaries that conserve angular momentum in the z direction, using only the TC interaction and global z fields. This includes controlled-Z, SWAP, and $\sqrt{i\text{SWAP}}$.

Figures

Figures reproduced from arXiv: 2506.03453 by the authors.

Figure 7.9
Figure 7.9. Toy model of a trapped ion: a single particle in a harmonic potential with two internal states, [PITH_FULL_IMAGE:figures/full_fig_p003_7_9.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Quantum circuit implementing the 2-qubit [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Under the action of [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. 3-qubit accidental symmetry: States are written in the form [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. 2-qubit example [PITH_FULL_IMAGE:figures/full_fig_p044_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p048_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p052_9.png]

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Works this paper leans on

28 extracted references · 27 canonical work pages · cited by 1 Pith paper

  1. [1]

    Matrix Elements ofH TC,J z,a †a The|j, m, k⟩basis is a simultaneous eigenbasis ofJz anda †a, with j, m, k a†a j,′ m′, k′ =k(δ j,j′δm,m′δk,k′) ⟨j, m, k|Jz |j′, m′, k′⟩=m(δ j,j′δm,m′δk,k′), Recall that HTC = gTC 2 nX i=1 σ(i) + a+σ (i) − a† =g TC (J+a+J −a†), (C9) For the remainder of this appendix, we set gTC = 1,(C10) to avoid unnecessary notational clutt...

    work page

  2. [2]

    In the|q, k⟩basis, ⟨q, k|HTC |q, k+ 1⟩=⟨q, k+ 1|H TC |q, k⟩= p (q−k)(k+ 1)(n−q+k+ 1),(C22) where0≤q≤ ∞andmax(0, q−n)≤k≤q−1

    Matrix Elements in the Symmetric Subspace For thej=n/2symmetric subspace, the nonzero matrix elements ofH TC are ⟨m, k|HTC |m−1, k+ 1⟩=⟨m−1, k+ 1|H TC |m, k⟩= p (n/2 +m)(n/2−m+ 1)(k+ 1), with−n/2 + 1≤m≤n/2and0≤k≤ ∞. In the|q, k⟩basis, ⟨q, k|HTC |q, k+ 1⟩=⟨q, k+ 1|H TC |q, k⟩= p (q−k)(k+ 1)(n−q+k+ 1),(C22) where0≤q≤ ∞andmax(0, q−n)≤k≤q−1. Also, q, k a†a q′...

    work page

  3. [3]

    Energy variance ofH TC Consider maximally-mixed state inC m(n,j) ⊗ Hq,j, the angular momentumjand chargeqsubspace in(C 2)⊗n ⊗ L2(R), τq,j := Πq,j Tr(Πq,j ) = Πq,j dn(q, j)m(n, j).(C25) 26 First, consider the symmetric subspace, which appears with multiplicity one, i.e.m(n, j=n/2) = 1, so Tr(π q,j=n/2 (HTC)) = Tr(HTCΠq,j=n/2 ). Then, using Eq.(C22), comput...

    work page

  4. [4]

    In the following, we determine the projection of the Lie algebra generated byH TC,J z, anda †aonto this center

    Projections of the Hamiltonians to the center of U(1)-invariant PI Hamiltonians (Charge Vectors) Recall that the center of the Lie algebragof all U(1)-invariant PI Hamiltonians is spanned byiΠ q,j. In the following, we determine the projection of the Lie algebra generated byH TC,J z, anda †aonto this center. (This projection, is also known as the charge v...

    work page

  5. [5]

    The first condition follows immediately from Theorem 13, and in the previous subsection, we proved that eachV∈ Vz satisfies the phase constraints in the second condition, Eq.(90)

    Proof of Theorem 8 We have already proved the necessity of the two conditions in Theorem 8. The first condition follows immediately from Theorem 13, and in the previous subsection, we proved that eachV∈ Vz satisfies the phase constraints in the second condition, Eq.(90). Conversely, here we argue that the two conditions of Theorem 8 are indeed sufficient:...

    work page

  6. [6]

    anharmonic

    Note: Constraint on the Relative Phases witha †a If we assume control of the intrinsic oscillator HamiltonianH osc =νa †ain addition toH TC andJ z, then the group of realizable unitaries, each of which can be written as a product V=e iθosc a†a LY ℓ=1 UT C(rℓ)Rz(sℓ), r ℓ, sℓ, θosc ∈[−T, T],(G22) 38 differs fromV z only by one additional free relative phase...

    work page

  7. [8]

    Consider the two-level subspace spanned by|0⟩ eff :=|00⟩ ⊗ |0⟩osc and|1⟩ eff :=|11⟩ ⊗ |2⟩osc, and let these states denote the±1eigenstates, respectively, of Pauli matrixσ z

    A geometric explanation ofF-gate Here, we present a geometric argument that explains why there existsϕ 0 andϕ 1 satisfying Eq.(K56). Consider the two-level subspace spanned by|0⟩ eff :=|00⟩ ⊗ |0⟩osc and|1⟩ eff :=|11⟩ ⊗ |2⟩osc, and let these states denote the±1eigenstates, respectively, of Pauli matrixσ z. Then, in this subspaceVandR z(θ)act as eV:=e iπ/2 ...

    work page

  8. [9]

    anharmonic

    Further Remarks on the Conserved OperatorS Recall from Section (VII) the definition S= n/2X j=jmin+1 j−1X j′=jmin S(j, j′),(E12) where forj ′ < j, we have defined S(j, j′) := j′ X m′=−j′ |j′, m′⟩⟨j, m′ −j+j ′| ⊗ |2j−j′ −m ′⟩⟨j′ −m ′|osc +H.C., (E13) It is worth noting that eachS(j, j ′)individually satisfies[H TC, S(j, j′)] = 0, for arbitraryj ′ < j. Howe...

    work page

Show all 28 references
  1. [10]

    Therefore, the unitaries that can be realized using the Hamiltonians HTC and HTC can also be realized usingH TC together with the identical single-qubit gatesS ⊗n

    Proof of Eq.(F3) Recall the decomposition (C2)⊗n ⊗ L2(R) = n/2M j=jmin ∞M q=n/2−j Cm(n,j) ⊗ Hq,j .(F8) 9 Note that, up to a global phase,exp(i π 2 Jz)is equal toS ⊗n, whereS=|0⟩⟨0|+i|1⟩⟨1|. Therefore, the unitaries that can be realized using the Hamiltonians HTC and HTC can al...

  2. [11]

    General version of Theorem 10 Here, we summarize and restate results originally from [61], which determine whether specific classes of Hamiltonians are (semi)-universal. Theorem 10, which we use to prove Theorem 8, is a special case of the following lemma: 11 Note that forj < ...

  3. [12]

    (Anharmonic system) At least one of the boundary energy gaps on the is different from all others, that is either (i)∆ 1 ̸= 0and|∆ y| ̸=|∆1|for ally >1, or (ii)∆ d−1 ̸= 0and|∆ y| ̸=|∆d−1|for ally < d−1

  4. [13]

    anharmonicity

    (Harmonic system) All energy gaps ofBare equal and nonzero, i.e.∆ y = ∆̸= 0for ally, and at least one of boundary second-order finite differences is different from all others, that is either (i)∆ 2ay ̸= ∆2a1 for ally >1 (ii)∆ 2ay ̸= ∆2ad−1 for ally < d−1. Here,Bis interpreted ...

  5. [14]

    From universality in each sector to full semi-universality The following lemma, quoted from [50], can be used to establish whether a subgroupT ⊆ V G ofG-invariant unitaries is semi-universal, that is whetherTcontains the commutator subgroup[V G,V G]. Lemma 12.LetV G be the set...

  6. [15]

    That is, SV U(1)×Sn Πqmax sym = [V U(1)×Sn ,V U(1)×Sn ]Πqmax sym = qmaxM q=0 SU Hq, j=n/2 ⊂ VzΠqmax sym .(G5) whereΠ qmax sym is the projector to Lqmax q=0 Hq, j=n/2

    Energy variance of TC Hamiltonian inside each sector with fixed angular momentumj (Proof of Theorem 6) Before proving Theorem 8, we first present the proof of the first statement in Theorem 6: for any integerq max ≥0,V z is semi-universal in the subspace Lqmax q=0 Hq, j=n/2, i...

  7. [16]

    Nevertheless, in this section, we show that aside from relative phases, pairwise correlations due to accidental symmetry are theonlysuch restrictions

    Accidental symmetry restricts full semi-universality in sectors with different angular momenta As explained in Section (VII) and Section (E), an accidental symmetry ofH TC prevents an extension of the semi-universality property to all sectorsH q,j. Nevertheless, in this sectio...

  8. [17]

    First, recall that any unitaryV∈ Vz realized by HamiltoniansH TC andJ z can be decomposed into a product, V= LY ℓ=1 e−iHTCrl e−iJzsℓ :r ℓ, sℓ ∈[−T, T](G14) for finiteLand timeT

    Relative phases Here, we show that any unitaryV∈ Vz satisfies the phase constraint Eq.(90) – this proves the necessity of the second condition in Theorem 8. First, recall that any unitaryV∈ Vz realized by HamiltoniansH TC andJ z can be decomposed into a product, V= LY ℓ=1 e−iH...

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    That is, we consider general initial states|ψ⟩ ⊗ |0⟩osc, where|ψ⟩ ∈C2 ⊗C 2 is an arbitrary state ofn= 2qubits

    Basics In this section, we study the dynamics of a pair of qubits, coupled to an oscillator initialized in the vacuum state, under Tavis- Cummings Hamiltonian and globalzfield. That is, we consider general initial states|ψ⟩ ⊗ |0⟩osc, where|ψ⟩ ∈C2 ⊗C 2 is an arbitrary state ofn...

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    :=e −iσx2π/ √ 3 denotes a rotation by angle4π/ √ 3about thex-axis on the Bloch sphere. Conjugating this gate by a well-chosen unitaryVwhich acts asV 1 in the charge-1sector, yields in the charge-1sector a rotation of fixed angle 4π/ √ 3about anarbitraryaxisˆn, and in charge-0a...

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    2-step”, “4-step

    Minimizing total interaction time of anA-gate Note that the interaction time of any particularA-gate (eqs. (K40) and (K43)) depends on the Euler angles in Eq.(K28). In fact, neither the 2-step decompositions of Theorem 16 nor the Euler decompositions in Eq.(K15) are unique. He...

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    Additionally, in theq= 2, j= 1sector,Fis Hermitian and thus is its own inverse,π 2,1(F) =π 2,1(F †) =π 2,1(F −1)

    Minimizing interaction times for any 2-qubit, PI,U(1)-invariant unitary In Section (IV), we showed that any two-qubit PI unitary which also preservesJ z can be implemented as U(θ, θ+, θ′, α) =eiαRz(θ′)A(θ, θ+)F Rz(θ)F .(K53) In fact, the job of theF-gate – swapping the states|...

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    Examples: CZ, SW AP, iSW AP, √ iSW AP For these four useful gates, the following table, Table V, gives the optimal interaction times using our constructions, as well as the corresponding parameters in terms of the decomposition U=e iαRz(θ′)A(θ, θ+)F Rz(θ)F , for those four gat...

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    Implementing five two-qubit gates: interaction times, and decompositions in terms of Eq.(25)

    := exp(i|Ψ+⟩⟨Ψ+|2π/ √ 3), 51 Gate Time (τ×g TC/2π) α θ′ θ θ+ Full Gate Sequence CZ ≈2.866 0 π/2 π/2 0 Rz( π 2 )A( π 2 ,0)F Rz( π 2 )F † SW AP ≈1.273 π 0 π π eiπA(π, π)Rz(π) iSW AP ≈2.546 −π/2 0 π/2 π e−iπ/2A( π 2 , π)F Rz( π 2 )F † √ iSW AP ≈2.688 −π/4 0 π/4 π/2 e−iπ/4A( π 4 ,...

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    Table VI lists the parameters of differentA-gates used to implement CZ, SW AP, iSW AP, √ iSW AP,Vqub.↔osc., andA(π 1,1(F †)

    Parameters ofA-gates used to implement CZ, SW AP, iSW AP, √ iSW AP,Vqub.↔osc. Table VI lists the parameters of differentA-gates used to implement CZ, SW AP, iSW AP, √ iSW AP,Vqub.↔osc., andA(π 1,1(F †). For eachA-gate, the four parametersθ 1(ˆn), θ2(ˆn)andθ 1( ˆm), θ2( ˆm)are ...

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    At the end of this section, we give an alternative proof using a geometric argument that explains why there should exist anglesϕ 0 andϕ 1 satisfying the above identity

    Consider one half of a a full period time evolution ofH TC in this sector, that is V:=π 2,1 VTC π/ √ 6 :=π 2,1 e−iHTCπ/ √ 6 =   1 3 0− q 8 9 0−1 0 − q 8 9 0− 1 3   .(K55) We prove that by combining three copies of the above3×3unitary with properz-rotations, we can real...

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Pith tools

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