REVIEW 3 major objections 4 minor 86 references
State preparation via measurement and feedback: pushing relations, state structures, and non-invertible symmetries
T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read The paper claims that the set of pushable virtual-bond defects, together with their pushing relations, classifies finite-depth measurement-feedback preparable states in one dimension and dictates their preparation circuits.
desk verdict A novel and plausible framework for 1D measurement-feedback state preparation, but the rank-1 checks that set the classification boundaries are algebraically wrong and need correction. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the pushable defect: a virtual-bond operator that satisfies Eq. (15) or (16), which by the MPS fundamental theorem is equivalent to an identity on the transfer matrix. From the target tensor $A$ and a set of pushable defects $\{V_t\}$, the paper builds an associated tensor $B$ whose $t$-th physical outcome is $AV_t$; $|A\rangle$ is one MF round from $|B\rangle$, so the whole preparation circuit is a chain $|A\rangle \leftarrow |B\rangle \leftarrow |C\rangle \leftarrow \cdots$ ending at an FDLU-preparable state. The classification of pushing relations (which defects go to identity, to $Z$, to $X$, or mix under pushing) is what selects the unitary gates, the number of rounds, and the associated duality symmetry.
What would settle it
A concrete falsifier: take the $Z\leftrightarrow X$ non-example tensor of Sec.~V D and check whether any non-identity virtual-bond operator satisfies the transfer-matrix identities (17) or (18); finding one would disprove the paper's expectation that this class has no pushable defects. Alternatively, exhibit an open-boundary MPS that is preparable for all $N$ by left-conditioned single-site measurement and feedback but admits no right-pushable defects; that would refute Theorem 1.
Extended reading notes
Core claim
For a translation-invariant MPS tensor $A$, a virtual-bond defect $V_t$ is right-pushable when $AV_t$ can be transformed into $AV'_t$ by a unitary on the physical leg, and the paper shows this is exactly a transfer-matrix identity. The paper proves (Lemma 3 and Theorem 1) that an open-boundary MPS is preparable from an associated state $|B\rangle$ by one round of single-site measurement and left-conditioned feedback if and only if $|B\rangle$ is built from $|A\rangle$ through right-pushable defects; iterating this construction gives the full preparation circuit whenever the target is OLMF-preparable. When all Pauli defects are pushable, the pushing relations decompose into classes $RIRI$, $RIRX$, $RZRX$, and $RILI$ (after blocking and gauge fixing), and the target is a product state up to a Tambara-Yamagami-type duality (KW, KT, or their composition) and a transversal unitary. When only Pauli-$Z$ defects push, classes such as $RI$--$RIRX$ and $RZ$--$RIRX$ arise and require one or two rounds of MF, with the two-round class connected to generalized cosine symmetries. The central assertion is that these defect sets and relations are not bookkeeping: they classify the states and dictate the circuits.
Load-bearing premise
The construction stands on the matrix-product-state fundamental theorem—after finite site blocking, equal transfer matrices force a unitary on the physical leg—together with the completeness theorem's reliance on open boundary conditions, invertible boundary tensors, and left-conditioned feedback; if any of these fails for a target state, the pushable-defect classification may not apply.
Editorial extensions
If this is right
- Every OLMF-preparable open-boundary 1D state has a preparation circuit explicitly constructed from its right-pushable defects; finding the circuit reduces to testing transfer-matrix identities, not searching over circuits.
- Fusion-measurement preparable states are precisely those whose MPS tensor carries a complete set of pushable Pauli defects, and their pushing relations identify the non-invertible duality (Kramers-Wannier, Kennedy-Tasaki, or a general Tambara-Yamagami duality) that maps a product state to them.
- Allowing defects to be pushed left as well as right does not enlarge the class of preparable states; the possible left/right relations collapse after blocking to the same four classes up to spatial inversion.
- States in the $RI$--$RIRX$ and $RZ$--$RILI$ classes are preparable by one round of measurement and feedback with non-transversal corrections, even though no complete set of transversally pushable Pauli defects exists.
- States in the $RZ$--$RIRX$ class can require two rounds of measurement and feedback and are related to product states by generalized cosine symmetries with fusion rule $L_{\alpha}L_{\alpha'}=L_{\alpha+\alpha'}+L_{\alpha-\alpha'}$.
Reading between the lines
- The completeness theorem suggests a concrete decision procedure for 1D OLMF preparability: enumerate Pauli defects, test the transfer-matrix identities, form the associated state, and check whether repeated blocking eventually makes its transfer matrix rank-1.
- If the paper's conjecture that the $Z\leftrightarrow X$ class has no pushable defects is correct, those states would be natural candidates for separating OLMF-preparable from more general OMF-preparable states, resolving the paper's open question about whether left-conditioned feedback can always be assumed.
- The equivalence noted in the outlook between $r$-round MF circuits and $r$ rounds of fan-out gates suggests a complexity-theoretic reading: the number of MF rounds in the defect-chain construction measures the adaptive depth needed to prepare a state, tying the classification to the magic hierarchy of shallow circuits.
- A higher-dimensional analogue would require a global notion of pushability through a tensor-network environment rather than a single transfer matrix; the 1D classes here provide a testbed for whether fusion-category data still classifies preparability in $d>1$.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. This paper develops a tensor-network framework for preparing one-dimensional matrix product states with constant-depth circuits that combine measurements and unitary feedback. The central objects are pushable virtual defects; the paper defines associated states |B>, shows that a 1-round MF circuit relating |B> to |A> exists when T_B is FDLU-preparable, classifies the possible pushing relations (RI, RZ, RIRX, RZRX, RILI, etc.), and claims completeness for open-boundary MPS under left-conditioned feedback (Theorem 1). It further connects the resulting classes to non-invertible dualities, including Tambara-Yamagami, Kennedy-Tasaki, and generalized cosine symmetries, with examples such as AKLT and symmetry-broken/cluster states.
Significance. If correct, the paper would supply a systematic, constructive classification of finite-depth MF-preparable states in one dimension and a concrete circuit-synthesis recipe, going beyond the fusion-measurement protocols of Refs. [19,20,28,29]. The derivations are analytical and self-contained rather than fitted; Theorem 1 is an attractive completeness statement; and the connection to non-invertible symmetries, together with explicit examples (AKLT via KT, Rep(D8) symmetry breaking, non-onsite symmetry breaking), makes the paper potentially important. However, several rank diagnostics used to decide FDLU-preparability of the associated states are algebraically incorrect, and these diagnostics are load-bearing for the 1-round versus 2-round classification.
major comments (3)
- [Sec. V B 1, Eq. (84)] The condition λ0λ3=λ1λ2 is asserted to imply that T_B=λ0I+λ1Z1+λ2Z2+λ3Z1Z2 is rank 1. This is false. In the eigenbasis of Z1,Z2 the eigenvalues are λ0+λ1+λ2+λ3, λ0+λ1−λ2−λ3, λ0−λ1+λ2−λ3, and λ0−λ1−λ2+λ3; rank 1 requires three of these to vanish. The product condition is only necessary, not sufficient. For example, (λ0,λ1,λ2,λ3)=(2,1,1,1/2) satisfies λ0λ3=λ1λ2 but gives eigenvalues 9/2, 3/2, 1/2, 1/2, so T_B has full rank. Consequently the RI–FDLU boundary includes states whose associated B is not FDLU-preparable, and the claimed 1-round circuit for those states is not justified.
- [Sec. V C 1, Eq. (91)] The same incorrect rank diagnostic is used for the RZ–FDLU class: the text states that when λ0λ3=λ1λ2 the transfer matrix T_B is of rank 1, but T_B in Eq. (91) has the same form λ0I+λ1Z1+λ2Z2+λ3Z1Z2 as in the RI case, so the same counterexample applies. The RZ–FDLU class is therefore misidentified: states with, e.g., (λ0,λ1,λ2,λ3)=(2,1,1,1/2) do not have an FDLU-preparable associated B and are not established to be 1-round MF-preparable by this argument.
- [Sec. V C 2, Eq. (96)] The claim that ⟨ψ_r|Z⊗I|ψ_r⟩=0 (equivalently λ3=0 in Eq. (91)) makes T_B=cI+c'Z⊗Z rank 1 is incorrect. The eigenvalues of cI+c'Z⊗Z are c+c' and c−c', each with multiplicity two, so the matrix has rank 1 only when c'=±c. When c'=0, T_B is proportional to the identity and has full rank. Thus the λ3=0 line is not the boundary at which B becomes FDLU-preparable; the correct boundary is λ3=±λ0. This shifts the division between RZ–FDLU (1 round) and RZ–RIRX (2 rounds), and the examples in Sec. VII C built on this boundary need to be reexamined.
minor comments (4)
- [Sec. III A, Eq. (19)] The step from equality of transfer matrices to a unitary relation between tensors uses the fundamental theorem for injective MPS; the required injectivity after site blocking should be stated explicitly at this point, since not every tensor in the paper is assumed to be in a canonical form.
- [Secs. II and V] Rank statements are sometimes made before site blocking, while Lemma 1 defines FDLU-preparability after finite blocking. Blocking does not remove nonzero eigenvalues of T_B, so this is not fatal, but the text should consistently indicate whether a rank statement is meant before or after blocking.
- [Sec. V D] The sentence 'we suspect that this state does not admit any pushable defect' is presented as a conjecture. If this is intended to support a no-go statement about finite-round MF preparation, it should be proved or explicitly labelled as an open problem.
- [General] The notation λ0λ3=λ1λ2 appears in Secs. V B 1 and V C 1 without an accompanying statement of which λ parameters are independent and which are constrained by complete positivity of the transfer matrix; adding the explicit four-eigenvalue form would make the error-corrected conditions much easier to verify.
Circularity Check
No significant circularity: the MF-preparability classification is constructive and self-contained; the flagged rank-one criteria are correctness risks, not circularity.
full rationale
The derivation chain is a constructive existence proof rather than a fit. Pushable defects are defined by local tensor identities (Eq. 15), the associated tensor B is built from A through Eq. 14, and FDLU-preparability is characterized by the rank-1 transfer-matrix criterion proved in Lemma 1; Lemmas 2 and 3 then construct a circuit rather than assuming one. Theorem 1's completeness direction reconstructs right-pushable defects from the feedback unitaries of an assumed OLMF circuit via the MPS fundamental theorem, so it proves an equivalence instead of restating a definition. No parameter is fitted to a target conclusion, and the KW/KT/TY/cosine connections are derived from the transfer-matrix identities (e.g., Eqs. 44, 51, 55, 98). The self-citations that occur (notably Ref. [65] in Sec. VII D) support the known symmetry-breaking properties of an illustrative example and are not load-bearing for the central classification. The algebraic rank-one conditions in Secs. V B 1, V C 1, and V C 2 have been challenged by diagonalization; if the challenge stands, the affected class boundaries would be incorrect, but an algebraic error is not circularity. The paper itself also flags a genuine limitation of Theorem 1 as a no-go tool in Sec. VIII (point 3), again a scope caveat rather than a circular step.
Assumptions & free parameters
assumptions (5)
- domain assumption Finite-depth MF circuits on 1D product states output MPSs with finite bond dimension; the local tensor is translation-invariant after site blocking.
- standard math Fundamental theorem of MPS: two tensors with identical transfer matrix and identical physical bond dimension are related by a unitary on the physical leg, after blocking to injective form.
- standard math A rank-1 transfer matrix after finite blocking characterizes FDLU-preparability.
- domain assumption For OBC completeness, boundary matrices L and R are invertible, the MPS is in minimal injective form, and feedback corrections are left-conditioned.
- standard math Any FDLU circuit can be reduced to a depth-2 brick-wall architecture via site blocking.
Cite this review
Pith. "Pith review of State preparation via measurement and feedback: pushing relations, state structures, and non-invertible symmetries." pith.science (2026). https://pith.science/paper/L7LMJGB5
@misc{pith2026260808821,
author = {Pith},
title = {Pith review of: State preparation via measurement and feedback: pushing relations, state structures, and non-invertible symmetries},
year = {2026},
howpublished = {\url{https://pith.science/paper/L7LMJGB5}},
note = {Machine review of arXiv:2608.08821}
}
abstract
Quantum circuits with measurements and unitary feedback (MF) can prepare long-range entangled states in constant depth, but a systematic construction of the MF preparation circuit for a given target state remains underexplored. We develop such a scheme for one-dimensional states, based on the notion of pushable defects: virtual-bond operators of a matrix product state that can be pushed through the tensor at the price of a physical feedback unitary. We show that the set of pushable defects, together with their pushing relations classifies finite-depth MF-preparable states and dictates their preparation circuits. To each class of the target state $|A\rangle$, we associate a state $|B\rangle$ from which $|A\rangle$ can be prepared using a 1-round MF circuit; in particular, $|A\rangle$ is preparable from a product state using a circuit with 1 round of MF whenever $|B\rangle$ is preparable by a finite-depth local unitary (FDLU) circuit. For a general target state, the scheme is obtained by iterating this procedure until the associated state is FDLU-preparable. For open-boundary matrix product states, the scheme is complete: it constructs a preparation circuit whenever finite-depth MF preparation with left-conditioned feedback corrections is possible. Pushable defects and pushing relations thus emerge as a unifying principle for quantum state preparation via measurements and feedback. This characterization further reveals an intrinsic connection between MF circuits and non-invertible symmetries: states with certain classes of pushing relations are related to a product state by Tambara-Yamagami duality operators, or by continuous cosine symmetry operators with fusion rules $L_{\alpha} L_{\alpha'} = L_{\alpha+\alpha'} + L_{\alpha-\alpha'}$, together with their generalizations up to (not necessarily transversal) gates.
Figures
Reference graph
Works this paper leans on
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[1]
This class containsRI–RIRIandRI–RILI, which correspond toλ 1 =λ 3 = 0 andλ 2 =λ 3 = 0, re- spectively
classRI–FDLU We useRI–FDLU to denote the class of states that sat- isfyZ7→IforA, and the associated state|B⟩is FDLU- preparable. This class containsRI–RIRIandRI–RILI, which correspond toλ 1 =λ 3 = 0 andλ 2 =λ 3 = 0, re- spectively. As long asλ 0λ3 =λ 1λ2,T B is of rank 1, and state|B⟩is FDLU-preparable according to Lemma
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[2]
Now that we have arbitrary compositions of generalized KW and KT for each subsystem, the dualities give rise to arbitrary bicharacter
with a diagonal bicharacter, whereas the KT duality realizes 17 another TY(Z 2 2), with an off-diagonal bicharacter. Now that we have arbitrary compositions of generalized KW and KT for each subsystem, the dualities give rise to arbitrary bicharacter. Lemma 6.In the decomposition of the virtual space into subspaces of dimensionsD 1, D2,· · ·, if the gener...
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[3]
(85) between|A⟩and|B⟩
Consequently,|A⟩is also FDLU-preparable using the relation presented in Eq. (85) between|A⟩and|B⟩. Using the representative state for a rank-1 transfer ma- trix derived in Eq. (11) of Sec. II, we present the general state structure of this class: · · · A A A · · · |+⟩ |+⟩ |+⟩ = · · · S S S UZ C UZ C UZ C U U · · · (86) whereUis a potential transversal gat...
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[4]
According to Eq
classRI–RIRX We useRI–RIRXto denote the pushing relation Z7→IforAandZ7→I,X7→XforB. According to Eq. (84), this requiresλ 1 =λ 2 = 0, and consequently this class of target state|A⟩has a transfer matrix in Eq. (83) or Eq. (I2) withλ 1 =λ 2 = 0. Recall that class RIRXis 1-round MF-preparable using fusion measure- ment, we obtain that the target state|A⟩is al...
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[5]
When λ0λ3 =λ 1λ2, this transfer matrix is of rank 1, then|B⟩ is FDLU-preparable and accordingly|A⟩is 1-round MF- preparable
classRZ–FDLU We useRZ–FDLU to denote the class of MPSs that satisfyZ7→ZforA, andBis FDLU-preparable. When λ0λ3 =λ 1λ2, this transfer matrix is of rank 1, then|B⟩ is FDLU-preparable and accordingly|A⟩is 1-round MF- preparable. Two special classes contained areRZ–RIRI andRZ–RILI, which correspond toλ 1 =λ 3 = 0 and λ2 =λ 3 = 0 in the transfer matrix in Eq. ...
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[6]
In particular, whenλ 1 =λ 2 = 0 in the transfer matrix in Eq
classRZ–RIRX We useRZ–RIRXto denote the pushing relationZ7→ ZforAandZ7→I,X7→XforB. In particular, whenλ 1 =λ 2 = 0 in the transfer matrix in Eq. (91),T B allows PauliXto be also pushable and therefore is in classRIRX. Consequently,|B⟩can be prepared using the 1-round MF-circuit that implements generalized KW, and|A⟩is 2-round MF-preparable. We derive a re...
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[7]
Therefore,|B⟩is FDLU-preparable and state|A⟩is 1-round MF-preparable. In Sec. VII C, we will discuss in detail specific examples of this class. The MPO in Eq. (94) describes a sequential circuit composed of controlled-Xgates and certain non-Clifford controlled-rotation gates. In particular, when onlyα 1 = αis non-zero, this MPO is related to the cosine sy...
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[8]
The MPS tensor is as follows: A= M Z C H |ψ⟩
classRIRX–FDLU The first family of states is given by KW·D M |ψ⟩⊗N , whereD M is an FDLU generated by tensorM. The MPS tensor is as follows: A= M Z C H |ψ⟩ . (102) The PauliZdefect in the top virtual subspace is pushed toI, and theXdefect in this subspace is pushed toX. The associated state|B⟩after a Bell projection in this subspace is given byD M |ψ⟩⊗N ....
Show all 86 references
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[9]
These states are 1-round MF-preparable, because KW can be implemented using 1-round MF cir- cuit as shown in Appendix D
classRIRI–RIRX The second family of states is given by FDLU· KW|ψ⟩ ⊗N . These states are 1-round MF-preparable, because KW can be implemented using 1-round MF cir- cuit as shown in Appendix D. Here, we demonstrate that these states are within classRIRI–RIRX. It suffices to dis...
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[10]
Following our previous ar- gument, after a blocking of finite sites, we can write the MPS tensor of these states as in classRIRI–RIRX–· · ·– RIRX–RIRI
and beyond We can continue to look at the family of states given by FDLU·KW· · ·KW·FDLU|ψ⟩⊗N , (104) for arbitrary FDLU circuits. Following our previous ar- gument, after a blocking of finite sites, we can write the MPS tensor of these states as in classRIRI–RIRX–· · ·– RIRX–R...
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[11]
It is realized in Ref
AKLT state The spin-1 AKLT state provides a paradigmatic ex- ample of symmetry-protected topological (SPT) or- der [60–62], and is also known as a resource state for measurement-based quantum computation (MBQC) [63]. It is realized in Ref. [64] that 1D AKLT state can be prepar...
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[12]
Focusing on the state preparation with one round of MF, in addition to the fusion measurement pro- tocol, we have studied the classRI–RIRX,RZ– FDLU,RIRI–RIRX, etc. The states we studied either differ from the fusion measurement prepara- ble state via an FDLU, or are preparable...
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[13]
In general, as long as the pushable defects form a complete or- thonormal basis, the target state can still be pre- pared using fusion measurement
In this work, we have only explored the classes of states with Pauli pushable defects. In general, as long as the pushable defects form a complete or- thonormal basis, the target state can still be pre- pared using fusion measurement. For pushable de- fects that cannot be brou...
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[14]
This result leads to a no-go theorem: an MPS with OBC cannot be pre- pared with left-conditioned feedback if it does not admit a set of pushable defects
For target MPSs defined on OBC, we have estab- lished in Theorem 1 that the preparation circuit for an OLMF-preparable target state can always be constructed under our scheme. This result leads to a no-go theorem: an MPS with OBC cannot be pre- pared with left-conditioned feed...
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[15]
Using the term in Ref
A closely related notion isshort-range nonstabi- lizerness (magic), which characterizes whether a state can be related to|0⟩ ⊗N by a Clifford circuit followed by a constant-depth or shallow-depth lo- cal circuit [66–68]. Using the term in Ref. [69], the short-range magic state...
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Some examples that can be prepared using fusion mea- surement protocols are studied [5, 50, 70]
The MF-preparation of states in the spatial dimen- sion greater than one is relatively less known. Some examples that can be prepared using fusion mea- surement protocols are studied [5, 50, 70]. It would be interesting to generalize our defect approaches to higher dimensions ...
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2026
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