The Bravyi–Poulin–Terhal tradeoff in two dimensions

Level 2 derivation: why bounded-range commuting constraints in an ordinary two-dimensional geometry force $$k d^2=O(n)$$ .

The memorable slogan is:

In two Euclidean dimensions, one distance- $$d$$ logical qubit consumes area of order $$d^2$$ .

This is not proved by drawing a logical string and counting its length. A general commuting-projector code need not be a stabilizer code and need not have string-shaped logical operators. The proof instead turns local correctability into an entropy statement:

a correctable square can be enlarged whenever its boundary is also correctable;
therefore every square whose side is a sufficiently small constant multiple of $$d$$ is correctable;
the lattice can be divided into two correctable checkerboards $$A,B$$ plus small corner patches $$C$$ ;
correctability of $$A$$ and $$B$$ forces all $$k$$ logical qubits through $$C$$ ;
the area of $$C$$ is $$O(n/d^2)$$ .

The third step is where two-dimensional geometry enters. The fourth is where quantum information differs from classical information: a region whose erasure is correctable has *negative* conditional entropy.

1. Precise statement

Put $$n$$ particles on a regular square lattice with open or periodic boundary conditions. Assume:

every site has Hilbert-space dimension at most a fixed $$q$$ ;
the codespace $\mathcal C$ is the common $$+1$$ eigenspace of pairwise commuting projectors $\Pi_a$ ;
the support of every $\Pi_a$ fits inside a $w\times w$ square;
$\dim\mathcal C=2^k$ ;
every erasure on fewer than $$d$$ sites is correctable.

Here $$w$$ and $$q$$ are independent of the lattice size. Then there is a constant $$c(w,q)$$ , independent of $$n,k,d$$ , such that

\boxed{k\,d^2\leq c(w,q)\,n.} \tag{1.1}

The theorem is often called a stabilizer-code bound because stabilizer codes are its most common application. The original result is stronger: commuting local projectors need not be Pauli projectors.

Three qualifications are load-bearing.

Local means Euclidean bounded range. A weight-six check may still be nonlocal if its six qubits separate by a distance growing with the code.
The constraints commute. The boundary factorization below uses commutativity, not merely a gapped local Hamiltonian.
The local dimension is bounded. Otherwise one site can hide a growing number of qubits and the entropy count $S(C)\leq |C|\log_2q$ has no size-independent coefficient.

If $$k=0$$ , (1.1) is immediate. The proof below therefore assumes $$k>0$$ . Cases with $$d$$ comparable to the fixed microscopic scale $$w^2$$ are also immediate from $k\leq n\log_2q$ ; the geometric argument handles the macroscopic-distance regime.

2. Dictionary: what “correctable” means

Let

\Pi=\prod_a\Pi_a

be the codespace projector. For a set of sites $$M$$ , write $\overline M$ for its complement.

2.1 Erasure criterion

A region $$M$$ is correctable when erasing all sites in $$M$$ can be reversed on every encoded state. The Knill–Laflamme condition for this known-location error is

\Pi O_M\Pi=c(O_M)\Pi \quad\text{for every operator }O_M \text{ supported in }M. \tag{2.1}

Why does (2.1) characterize erasure? Choose an operator basis $\{E_\alpha\}$ on $$M$$ . Erasure has error products $E_\alpha^\dagger E_\beta$ , and these span the full operator algebra on $$M$$ . The ordinary Knill–Laflamme equations for all pairs $\alpha,\beta$ are therefore exactly (2.1).

Two consequences will be used repeatedly.

Any subset of a correctable region is correctable.
Every region with fewer than $$d$$ sites is correctable, by the definition of distance used in the theorem.

For a Pauli stabilizer code, (2.1) says that a region contains no nontrivial logical Pauli. Thus this erasure distance agrees with the usual minimum weight of a nontrivial logical Pauli.

2.2 The entropy identity

Let $\rho$ be any encoded state. Correctability of $$M$$ implies

S(M\mid\overline M)=-S(M), \tag{2.2}

where logarithms are base two.

Here is the information-theoretic content of the equation. Purify the logical input with a reference $$R$$ , encode it into $M\overline M$ , and apply (2.1). No observable on $$M$$ depends on the logical input, including its off-diagonal coherences, so

\rho_{RM}=\rho_R\otimes\rho_M .

Equivalently $$I(R:M)=0$$ . The global state on $RM\overline M$ is pure, hence

\begin{aligned} 0=I(R:M) &=S(R)+S(M)-S(RM)\\ &=S(M\overline M)+S(M)-S(\overline M). \end{aligned}

Rearranging gives (2.2). The negative sign is not a formal curiosity: $\overline M$ can reconstruct the quantum information lost from $$M$$ , so the two regions share the entanglement needed to undo that loss.

For the maximally mixed encoded state

\rho_*=\frac{\Pi}{2^k},

the entropy of the full lattice is

S(\Lambda)=k. \tag{2.3}

Equations (2.2) and (2.3) will convert geometry into a bound on $$k$$ .

2.3 Internal and external boundaries

The external boundary $\partial_+M\subseteq\overline M$ contains every outside site belonging to a projector that also touches $$M$$ . The internal boundary is

\partial_-M=\partial_+(\overline M),

and $\partial M=\partial_-M\cup\partial_+M$ .

For an $R\times R$ square on a square lattice,

|\partial M|\leq c_\square\,w(R+w), \tag{2.4}

where $c_\square$ is a geometry-only constant. Equation (2.4) is simply a perimeter count: only a collar of thickness $$O(w)$$ can participate in a constraint crossing the cut.

3. Why commuting projectors can be split at a boundary

The proof must do more than clean Pauli operators. It must disentangle an arbitrary commuting-projector codespace at the boundary of a correctable region.

3.1 Boundary-algebra decomposition

Suppose two commuting projectors act on $$AB$$ and $$BC$$ :

P_{AB}Q_{BC}=Q_{BC}P_{AB}.

There is a decomposition

\mathcal H_B= \bigoplus_x \mathcal H_{B_x^L}\otimes\mathcal H_{B_x^R} \tag{3.1}

such that, in every sector $$x$$ , the $$AB$$ projector acts only on $$AB_x^L$$ and the $$BC$$ projector acts only on $$B_x^RC$$ .

To see the mechanism, take operator-Schmidt expansions

P_{AB}=\sum_i A_i\otimes L_i,\qquad Q_{BC}=\sum_j R_j\otimes C_j

with linearly independent outer operators. Commutativity forces $$[L_i,R_j]=0$$ for every $$i,j$$ . The finite-dimensional algebras $\mathcal L=\operatorname{alg}\{L_i\}$ and $\mathcal R=\operatorname{alg}\{R_j\}$ therefore commute.

Decompose the center of $\mathcal L$ into minimal central projectors. Inside one central block, a finite-dimensional factor is a full matrix algebra on one tensor factor with an identity on its multiplicity factor:

\mathcal L_x\simeq \mathsf M(\mathcal H_{B_x^L})\otimes I_{B_x^R}.

Every operator commuting with $\mathcal L_x$ acts on the other factor, so

\mathcal R_x\subseteq I_{B_x^L}\otimes\mathsf M(\mathcal H_{B_x^R}).

This proves (3.1). The label $$x$$ is a classical boundary sector; the left and right quantum degrees of freedom separate inside each sector.

For many commuting constraints, group the product of all constraints meeting the left side into $P_{AB}$ and the product of the rest into $Q_{BC}$ . Their interaction algebras obey the same decomposition.

3.2 Boundary Disentangling Lemma

Lemma. Let $$M$$ and its external boundary $\partial_+M$ both be correctable. Then a unitary supported on $\partial M$ transforms the codespace projector into

U_{\partial M}\Pi U_{\partial M}^\dagger =|\phi_M\rangle\langle\phi_M|\otimes\Pi_{\overline M} \tag{3.2}

for a fixed pure state $|\phi_M\rangle$ independent of the encoded state.

Proof. Partition the lattice as

\begin{aligned} A&=M\setminus\partial_-M,& B&=\partial_-M,\\ C&=\partial_+M,& D&=\overline M\setminus\partial_+M. \end{aligned}

Thus $$M=AB$$ , $\overline M=CD$ , and all communication between the two interiors passes through $$BC$$ .

First split $$C$$ . Group the commuting constraints into those supported on $$ABC$$ and those supported on $$CD$$ . Applying (3.1) to their interaction algebras gives

\Pi=\bigoplus_x \Pi^{(x)}_{AB\,C_x^L}\otimes \Pi^{(x)}_{C_x^R D}. \tag{3.3}

If two nonzero sectors $$x$$ occurred, measuring the sector label on $$C$$ would distinguish orthogonal code states. That contradicts correctability of $C\subseteq\partial_+M$ . Hence only one sector occurs, and a unitary $$U_C$$ exposes a tensor split $$C=C^LC^R$$ .

Apply the same argument to the internal boundary $$B$$ , grouping constraints on $$AB$$ against those on $$BCD$$ . Because $B\subseteq M$ is correctable, only one boundary sector occurs. After a unitary $$U_B$$ , the projector has the product form

\Pi'= \Pi_{AB^L}\otimes \Theta_{B^R C^L}\otimes \Pi_{C^R D}. \tag{3.4}

Correctability of $$M$$ forces $\Pi_{AB^L}$ to have rank one. Otherwise two orthogonal states in that factor could be distinguished by an observable supported in $$M$$ .

The middle projector $\Theta_{B^RC^L}$ also has rank one. If it encoded a nontrivial system, correctability of $B^R\subseteq M$ would make that system recoverable from $$C^L$$ , while correctability of $C^L\subseteq\partial_+M$ would make the same system recoverable from $$B^R$$ . Two disjoint systems would then each contain the full unknown quantum state, contradicting no-cloning.

A final unitary on $$B^RC^L$$ maps its fixed pure state into the $$M$$ factor. Together with $$U_B$$ and $$U_C$$ , it gives (3.2). $\square$

The unitary $U_{\partial M}$ need not be a geometrically local circuit inside the boundary. The theorem needs only its support, not a depth bound.

3.3 Growth Lemma: a correctable boundary lets a region expand

Lemma. Write the lattice as $$ABCD$$ , with $$M=AB$$ . Suppose:

$$M$$ is correctable;
$$BC$$ is correctable;
every constraint crossing the boundary of $$M$$ is supported in $$BC$$ .

Then $ABC=M\cup C$ is correctable.

Proof. Since $\partial_+M\subseteq C\subseteq BC$ , both $$M$$ and its external boundary are correctable. Apply (3.2). After the boundary unitary, every encoded state has the form

\eta_{AB}\otimes\eta_{CD}, \tag{3.5}

where $\eta_{AB}$ is a known pure state independent of the logical input.

Consider the error that first applies the boundary unitary from (3.2) and then erases $$BC$$ . Its support lies in the correctable region $$BC$$ , so it has a recovery map. Its surviving state is

\eta_A\otimes\rho_D. \tag{3.6}

The factor $\eta_A$ is fixed and can be prepared for free. Therefore $\rho_D$ alone determines and reconstructs the original encoded state. Erasure of $$ABC$$ is correctable. $\square$

This is the holographic step: once the interior is known to be correctable, only a boundary collar can obstruct enlarging it.

3.4 Separated-Union Lemma

Lemma. Let $M_1,\ldots,M_t$ be correctable regions such that no local projector overlaps two of them. If every external boundary $\partial_+M_i$ is correctable, then

M_1\cup\cdots\cup M_t

is correctable.

Proof. It is enough to verify (2.1). Expand an arbitrary operator on the union as a sum of tensor products $O_{M_1}\otimes\cdots\otimes O_{M_t}$ .

Apply the factorization from the proof of the Disentangling Lemma to $$M_1$$ . No constraint touching $$M_1$$ reaches any other $$M_j$$ , so

\Pi (O_{M_1}\otimes O_{M_2}\otimes\cdots\otimes O_{M_t}) \Pi =f_1(O_{M_1}) \Pi(O_{M_2}\otimes\cdots\otimes O_{M_t})\Pi. \tag{3.7}

Repeat for $M_2,\ldots,M_t$ . The result is

\Pi O\Pi=c(O)\Pi,

which is (2.1) for the union. $\square$

This is the first point where bounded interaction range matters. Weight alone is insufficient: one bounded-weight projector could touch two distant blocks and prevent (3.7).

4. Holographic correctability: side length is linear in distance

Let $$R_*$$ be the largest integer such that every $R_*\times R_*$ square is correctable. Such a maximum exists for $$k>0$$ , because the full lattice cannot be erased and recovered.

Distance alone gives only $R_*=\Omega(\sqrt d)$ : a square with fewer than $$d$$ sites is correctable. Locality improves this to

R_*=\Omega(d/w). \tag{4.1}

Take any correctable $R_*\times R_*$ square $$M$$ . Let $$B$$ be an internal collar and $$C$$ an external collar, each of thickness sufficient to contain the support of every constraint crossing the cut. Then $\partial M\subseteq BC$ , and (2.4) gives

|BC|\leq c_\square w(R_*+w). \tag{4.2}

Suppose the right-hand side were smaller than $$d$$ . The collar $$BC$$ would be correctable. The Growth Lemma would make $M\cup C$ , a strictly larger square, correctable. The construction works at every position, so it would contradict maximality of $$R_*$$ .

Therefore

d\leq c_\square w(R_*+w). \tag{4.3}

When $$d$$ is larger than a fixed multiple of $$w^2$$ , (4.3) gives (4.1). When it is not, the final theorem follows from the trivial entropy bound $k\leq n\log_2q$ after enlarging $$c(w,q)$$ .

Now choose a smaller mesoscopic length

\ell=\left\lfloor\alpha\,\frac d w\right\rfloor, \tag{4.4}

where the geometry-only constant $\alpha>0$ is small enough that:

$\ell\leq R_*$ , so every $\ell\times\ell$ square is correctable;
every width- $$O(w)$$ boundary collar of such a square has fewer than $$d$$ sites and is therefore correctable.

This is the exact content hidden by the phrase “a box of side proportional to $$d$$ is correctable.” The proportionality constant depends on the interaction range because the information bottleneck is the perimeter $O(w\ell)$ .

5. The checkerboard partition

Tile the lattice by $\ell\times\ell$ squares and color the tiles black and white in a checkerboard pattern.

$$A$$ is the union of the black tiles after shaving an $$O(w)$$ -sized patch from every corner.
$$B$$ is the analogous union of the white tiles.
$$C$$ is the union of all shaved corner patches.

The corner patches have two jobs. Black tiles meet other black tiles only at corners, as do white tiles. Removing a collar wider than the interaction range ensures that no projector touches two components of $$A$$ , and no projector touches two components of $$B$$ .

Each component of $$A$$ or $$B$$ is a subset of an $\ell\times\ell$ correctable square. Its external boundary is correctable by the choice of $\ell$ . The Separated-Union Lemma therefore gives:

A\text{ and }B\text{ are correctable}. \tag{5.1}

There are $O(n/\ell^2)$ tile corners, and each removed patch contains $$O(w^2)$$ sites. Boundary fragments at the edge of an open lattice change only the geometry constant. Hence

|C|\leq c_C\,\frac{n w^2}{\ell^2}. \tag{5.2}

The diagram to keep in mind is not “large boxes separated by wide corridors.” Wide corridors would occupy $O(nw/\ell)$ sites and lose one power of $$d$$ . The checkerboard needs only codimension-two intersections: in two dimensions, those are isolated corner patches.

6. The entropy calculation

Evaluate the maximally mixed code state $\rho_*=\Pi/2^k$ on the partition $\Lambda=ABC$ .

Correctability of $$A$$ and (2.2) give

\begin{aligned} k &=S(ABC)\\ &=S(BC)+S(A\mid BC)\\ &=S(BC)-S(A)\\ &\leq S(B)+S(C)-S(A). \end{aligned} \tag{6.1}

The last line is subadditivity. Interchanging $$A$$ and $$B$$ gives

k\leq S(A)+S(C)-S(B). \tag{6.2}

Add (6.1) and (6.2). The unknown entropies of the two large checkerboards cancel:

k\leq S(C). \tag{6.3}

Each site has dimension at most $$q$$ , so

S(C)\leq |C|\log_2q. \tag{6.4}

Substitute (5.2) and $\ell=\Theta(d/w)$ :

\begin{aligned} k &\leq c_C\log_2q\,\frac{n w^2}{\ell^2}\\ &\leq c'(w,q)\,\frac{n}{d^2}. \end{aligned}

Equivalently,

\boxed{k d^2\leq c'(w,q)n}.

That completes the proof.

The flow of logical information is now visible. Both extensive regions $$A$$ and $$B$$ may be erased. Quantum information cannot reside in either one, yet together they cover almost the whole lattice. Their two entropy identities force the logical entropy into the sparse intersection skeleton $$C$$ , and geometry makes that skeleton too small to carry more than $$O(n/d^2)$$ qubits.

7. Consequences and saturation

7.1 Fixed $$k$$

For any family encoding at least one logical qubit,

d=O(\sqrt n).

This is why a two-dimensional surface-code patch with $n=\Theta(d^2)$ has the optimal distance exponent. The theorem does not select the surface code or its constant prefactor; it says no code under the same hypotheses can improve the exponent.

7.2 Nonzero rate

If $$k/n$$ stays above a positive constant, then

\[ d=O(1). \]

Thus a two-dimensional bounded-range commuting-projector family cannot simultaneously have constant rate and growing distance.

7.3 Tightness

One planar surface-code patch has

k=1,\qquad n=\Theta(d^2).

Placing $$k$$ independent patches side by side gives

n=\Theta(kd^2).

The BPT scaling is therefore tight up to a constant depending on layout and microscopic locality. Rotating a patch or changing its boundary convention can improve that constant, but cannot change the exponent while the theorem's hypotheses remain true.

7.4 Higher dimensions

The same proof in $$D$$ Euclidean dimensions uses a codimension-two skeleton:

the sparse region has size $O(n/\ell^2)$ ;
a cube of side $\ell$ has boundary size $O(\ell^{D-1})$ ;
holographic correctability permits $\ell^{D-1}=O(d)$ .

Consequently,

k\leq c_D\,\frac{n}{d^{\,2/(D-1)}}. \tag{7.1}

For $$D=2$$ , (7.1) is the bound proved above. The original paper explicitly notes that its higher-dimensional exponent need not be tight.

7.5 Why the classical companion has a different exponent

The same paper proves that a two-dimensional classical code specified by bounded-range constraints obeys

k\sqrt d=O(n). \tag{7.2}

The shorter proof exposes the quantum difference. Choose separated square blocks $B_1,\ldots,B_t$ , each containing fewer than $$d$$ sites and having side $\Theta(\sqrt d)$ . Let $B=\bigcup_iB_i$ , and let $$A$$ be the width- $$O(w)$$ grid between them. Then

|A|=O(nw/\sqrt d). \tag{7.3}

Suppose two classical codewords $$x,y$$ agree on $$A$$ . If they differ inside some block $$B_i$$ , splice a new assignment $$z$$ : use $$x$$ on $$B_i$$ and $$y$$ everywhere else. Every local constraint sees at most one block, and constraints meeting $$B_i$$ see the same boundary values from $$x$$ and $$y$$ , so $$z$$ is another codeword. But $$z$$ and $$y$$ differ on fewer than $$d$$ sites, contradicting distance. Therefore agreement on $$A$$ determines the whole codeword.

The restriction map from codewords to configurations on $$A$$ is injective. For bounded local alphabet size, the number of encoded bits satisfies

k\leq c_q|A|,

which together with (7.3) gives (7.2).

Classically, one thin grid is enough because a codeword can be copied and compared. Quantumly, no-cloning and entanglement replace unique extension by the negative conditional-entropy identity. Two correctable checkerboards are needed, and their entropy cancellation leaves a codimension-two set. That is why the quantum tradeoff contains $$d^2$$ , while the classical companion contains only $\sqrt d$ .

8. What does and does not evade the theorem

Change	Covered?	Reason
Replace square plaquettes by another bounded-density 2D tiling	Yes	Only bounded Euclidean geometry and bounded interaction range enter
Use qudits of a fixed dimension	Yes	The coefficient acquires a factor depending on $\log_2 q$
Use a Pauli stabilizer code with local generators	Yes	It is a special case of commuting local projectors
Keep checks low-weight but connect sites a growing Euclidean distance apart	No	Weight does not bound $$w$$ ; the separated-union step fails
Embed the Tanner graph in two layers with long bridges	Not with a size-independent $$w$$	The relevant metric is the physical Euclidean layout, including bridges
Let the interaction range grow with $$n$$	No uniform asymptotic bound	The hidden coefficient depends on $$w$$
Use noncommuting local gauge checks of a subsystem code	Not by the original theorem	The commuting-projector boundary decomposition is unavailable
Move to a non-Euclidean family of graphs	Not as stated	The checkerboard and codimension-two counting use ordinary Euclidean growth
Move to $$D>2$$	A different tradeoff applies	Use (7.1), not the two-dimensional exponent
Improve a finite block's value of $$kd^2/n$$	Possibly	The theorem fixes asymptotic scaling, not a universal finite-size constant

Two common confusions deserve emphasis.

LDPC is not geometric locality. LDPC bounds how many qubits a check touches and how many checks touch a qubit. BPT additionally requires those qubits to lie in a bounded Euclidean neighborhood. A family may be qLDPC and still lie outside BPT.

BPT is not a threshold theorem. It constrains the static code parameters $$(n,k,d)$$ . It says nothing by itself about syndrome circuits, decoder runtime, noise thresholds, or logical error rates.

9. BPT versus nearby no-go theorems

Result	Resource constrained	Typical conclusion in 2D
Bravyi–Poulin–Terhal	Static information density under local commuting constraints	$$kd^2=O(n)$$
Bravyi–König	Logical gates implemented by bounded-depth local circuits	Only Clifford logical gates
Bravyi–Terhal	Thermal energy barrier of 2D local stabilizer Hamiltonians	No growing stabilizer energy barrier
Eastin–Knill	Exactly transversal gates on a finite-dimensional exact code	No universal transversal gate set

These statements share examples but not proofs. BPT uses correctable-region entropy and a geometric partition. It does not follow from the Bravyi–König commutator argument, and neither theorem says that a surface code is self-correcting.

10. Audit questions

Q1. Why does distance first suggest only a $\sqrt d$ -sized correctable square, yet the proof obtains a $$d$$ -sized square?

A. Distance alone counts the square's area: fewer than $$d$$ sites gives side $O(\sqrt d)$ . Commuting locality changes the bottleneck from area to boundary. The Growth Lemma says a correctable interior can keep expanding until its width- $$w$$ boundary collar has $$O(d)$$ sites, which occurs at side $$O(d/w)$$ .

Q2. Where exactly is commutativity used?

A. It makes the left and right interaction algebras on a boundary commute. Their direct-sum tensor-product decomposition gives the Disentangling Lemma and the Separated-Union Lemma. The final entropy inequalities alone do not use commutativity.

Q3. Why are two correctable checkerboards needed?

A. One correctable region gives $k\leq S(B)+S(C)-S(A)$ , which still contains the extensive difference $$S(B)-S(A)$$ . Applying the same identity to $$B$$ gives the opposite difference; adding cancels both and leaves $k\leq S(C)$ .

Q4. Why are the removed regions corner patches rather than corridors?

A. Same-colored checkerboard tiles meet only at corners. Removing bounded patches there separates equal colors. There are $O(n/\ell^2)$ corners, so the remainder has the $1/\ell^2$ density needed for $$1/d^2$$ . Width- $$w$$ corridors would have density $O(w/\ell)$ and lose one power.

Q5. Does a finite block with $$kd^2/n>1$$ violate BPT?

A. No. The constant in $$O(n)$$ is not universally one, and a single finite block cannot violate an asymptotic statement. To apply BPT to a family, $$q$$ , the Euclidean interaction range, and the geometric bounded-density constants must all remain fixed as $$n$$ grows.

Q6. Why does the theorem include non-Pauli commuting-projector codes but exclude generic subsystem codes?

A. Pauli algebra is not used. Pairwise commutativity of the defining projectors is. Local subsystem gauge generators generally do not commute, and their commuting stabilizers may be geometrically nonlocal, so the boundary-algebra step does not automatically apply.

11. Source and citation audit

Primary source, fetch-verified on 2026-07-28:

Sergey Bravyi, David Poulin, and Barbara Terhal, “Tradeoffs for reliable quantum information storage in 2D systems,” arXiv:0909.5200, submitted 28 September 2009; Physical Review Letters 104, 050503 (2010).

Boundary-algebra source, also fetch-verified on 2026-07-28:

Sergey Bravyi and Mikhail Vyalyi, “Commutative version of the $$k$$ -local Hamiltonian problem and common eigenspace problem,” arXiv:quant-ph/0308021; *Quantum Information & Computation* 5, 187–215 (2005).

The precise hypotheses, erasure definition of distance, entropy identity, Disentangling Lemma, Growth Lemma, separated-union argument, checkerboard partition, quantum bound, classical contrast, and $$D$$ -dimensional generalization trace to the primary BPT source. This page expands the boundary-algebra step, correctability criterion, entropy cancellation, and constant-scale choices so that the cited paper supplies provenance rather than a missing proof.

Atlas citation note: atlas/surface/card.yaml uses the bibkey bravyi2010tradeoffs, while docs/lit/surface/surface-foundations.md uses bravyipoulinterhal2010tradeoffs for the same fetch-verified paper. T5 permits new files only, so this page records the mismatch instead of changing either out-of-scope file. The planned reference-registry task should choose one key and enforce it repository-wide.

Result	Resource constrained	Typical conclusion in 2D
Bravyi–Poulin–Terhal	Static information density under local commuting constraints	$\(kd^2=O(n)\)$
Bravyi–König	Logical gates implemented by bounded-depth local circuits	Only Clifford logical gates
Bravyi–Terhal	Thermal energy barrier of 2D local stabilizer Hamiltonians	No growing stabilizer energy barrier
Eastin–Knill	Exactly transversal gates on a finite-dimensional exact code	No universal transversal gate set