No-Go Theorem: Degenerate Ground Spaces of Gapped Hamiltonians are Constant-Distance AQEDCare Constant-Distance AQEDC

QUANTUM ERROR-DETECTION AT LOW ENERGIES

2.5 No-Go Theorem: Degenerate Ground Spaces of Gapped Hamiltonians are Constant-Distance AQEDCare Constant-Distance AQEDC

The main result of this section is the following upper bound on the matrix elements of geometrically𝑑-local operators with respect to two MPS.

Theorem 2.4.5. Let|Ψ₁i= |Ψ(𝐴

1, 𝑋

1, 𝑛)i,|Ψ₂i= |Ψ(𝐴

2, 𝑋

2, 𝑛)i ∈ (C^p)^⊗𝑛be two MPS with bond dimensions𝐷

1and𝐷

2, where

𝑋_𝑗 =|𝜑_𝑗ih𝜓_𝑗|, with k𝜑_𝑗k = k𝜓_𝑗k =1 for 𝑗 =1,2, are rank-one operators. Let𝐸 =𝐸(𝐴

1, 𝐴

2) ∈ B (C^𝐷1 ⊗C^𝐷2) denote the combined transfer operator defined by the MPS tensors 𝐴

1 and 𝐴

2, ℎ^∗

𝑗 the size of the largest Jordan block of𝐸_𝑗 =𝐸(𝐴_𝑗) for 𝑗 =1,2, andℎ^∗the size of the largest Jordan block of 𝐸 = 𝐸(𝐴

1, 𝐴

2). Assume that the spectral radii 𝜌(𝐸), 𝜌(𝐸

1), and 𝜌(𝐸

2) are contained in [0,1]. Then, for any𝐹 ∈ B ( (C^p)^⊗𝑑), we have

|hΨ₁| (𝐹⊗ 𝐼₍

C^p)^{⊗𝑛−𝑑}) |Ψ₂i| ≤ 16· k𝐹k ·𝑑^(ℎ

∗ 1+ℎ^∗

2−2)/2(𝑛−𝑑)^ℎ∗−1 for𝑑 𝐷

1, 𝐷

2and (𝑛−𝑑) 𝐷

1𝐷

2. Proof. The matrix elements𝛼= hΨ₁| (𝐹 ⊗𝐼₍

C^p)^{⊗𝑛−𝑑}) |Ψ₂iof interest can be written as

𝛼= (h𝜓

1| ⊗ h𝜓

2|)𝐸_𝐹𝐸^𝑛−𝑑(|𝜑

1i ⊗ |𝜑

2i) . By the Cauchy-Schwarz inequality, we have

|𝛼| ≤ k𝐸^†

𝐹(|𝜓

1i ⊗ |𝜓

2i) k · k𝐸^𝑛−𝑑(|𝜑

1i ⊗ |𝜑

2i k

≤ k𝐹k

√︃

k𝐸^𝑑

1k · k𝐸^𝑑

2k · k𝐸^𝑛−𝑑k,

by the definition of the operator norm and Lemma 2.4.4. Then, the claim follows from Lemma 2.4.2 (i), which provides the bounds

k𝐸^𝑑

𝑗k ≤ 4𝑑^ℎ

∗ 𝑗−1

for 𝑗 =1,2, k𝐸^𝑛−𝑑k ≤ 4(𝑛−𝑑)^ℎ∗−1

by our assumptions : 𝜌(𝐸_𝑗) ∈ [0,1],𝜌(𝐸) ∈ [0,1], and𝑑 𝐷_𝑗 ≥ ℎ^∗

𝑗 for 𝑗 =1,2, as well as𝑛−𝑑 𝐷

1𝐷

2 ≥ ℎ^∗.

2.5 No-Go Theorem: Degenerate Ground Spaces of Gapped Hamiltonians

than constant. We prove this result by employing the necessary condition for approximate error-detection from Lemma 2.3.6 for the code subspaces generated by varying the boundary conditions of an (open-boundary) injective MPS. Note that, given a translation invariant MPS with periodic boundary conditions and bond dimension𝐷, there exists a local gapped Hamiltonian, called the parent Hamiltonian, with a unique ground state being the MPS [28].

We need the following bounds which follow from the orthogonality and normaliza- tion of states in such codes.

Lemma 2.5.1. Let𝐴be the MPS tensor of an injective MPS with bond dimension𝐷, and let 𝑋 , 𝑌 ∈ B (C^𝐷) be such that the states |Ψ𝑋i = |Ψ(𝐴, 𝑋 , 𝑛)i and |Ψ𝑌i =

|Ψ(𝐴, 𝑌 , 𝑛)i are normalized and orthogonal. Let us write the transfer operator as 𝐸 =|𝐼iihhΛ| ⊕𝐸˜ (cf. Equation(2.4.2)). Assume𝑛 𝐷. Then

(i) The Frobenius norm of𝑋 (and similarly the norm of𝑌) is bounded by k𝑋k𝐹 =𝑂(1) .

(ii) We have

|hhΛ| (𝑋 ⊗𝑌) |𝐼ii| =𝑂(𝜆

𝑛/2 2 ),

|hhΛ| (𝑌 ⊗ 𝑋) |𝐼ii| =𝑂(𝜆

𝑛/2 2 ). In the following proofs, we repeatedly use the inequality

|tr(𝑀

1. . . 𝑀_𝑘) | ≤ k𝑀

1k𝐹 · k𝑀

2k𝐹· · · k𝑀_𝑘k𝐹 (2.30) for 𝐷× 𝐷-matrices {𝑀_𝑗}^𝑘

𝑗=1. Note that the inequality (2.5) is simply the Cauchy- Schwarz inequality for𝑘 =2. For 𝑘 >2, the inequality follows from the inequality for𝑘 =2 and the submultiplicativity of the Frobenius-norm because

|tr(𝑀

1. . . 𝑀_𝑘) | ≤ k𝑀

1k𝐹 · k𝑀

2· · ·𝑀_𝑘k𝐹 ≤ k𝑀

1k𝐹 · k𝑀

2k𝐹· · · k𝑀_𝑘k𝐹 . Proof. The proof of (i) follows from the fact that the stateΨ𝑋 is normalized, i.e.,

1=kΨ𝑋k²

=tr

𝐸^𝑛(𝑋 ⊗ 𝑋)

=tr

|𝐼iihhΛ| (𝑋 ⊗ 𝑋)

+tr(𝐸˜^𝑛(𝑋 ⊗ 𝑋))

=tr(Λ𝑋 𝑋^†) +tr(𝐸˜^𝑛(𝑋 ⊗ 𝑋))

≥ 𝜆

min(Λ) · k𝑋k²_𝐹+tr(𝐸˜^𝑛(𝑋 ⊗ 𝑋)),

where𝜆

min(Λ) denotes the smallest eigenvalue of Λ, and we make use of the fact that𝑋 𝑋^†is positive with trace tr(𝑋 𝑋^†) = k𝑋k²

𝐹. Since

|tr(𝐸˜^𝑛(𝑋 ⊗ 𝑋)) | ≤ k𝐸˜^𝑛k𝐹 · k𝑋 ⊗ 𝑋k𝐹 ≤ 𝜆^𝑛/2

2 k𝑋k²_𝐹 for𝑛 𝐷 by (2.5) and (2.4.2), we conclude

k𝑋k²_𝐹 ≤

𝜆min(Λ) −𝜆^𝑛/2

2

−1

=𝜆

min(Λ)⁻¹(1+𝑂(𝜆^𝑛/2

2 )). Then the claim (i) follows since𝜆

min(Λ)⁻¹is a constant.

Now, consider the first inequality in (ii) (the bound for |hhΛ| (𝑌 ⊗ 𝑋) |𝐼ii| is shown analogously). Using the orthogonality of the states|Ψ𝑋iand|Ψ𝑌i, we obtain

0=hΨ𝑋|Ψ𝑌i=tr

𝐸^𝑛(𝑋 ⊗𝑌)

=tr

(|𝐼iihhΛ| +𝐸˜^𝑛) (𝑋 ⊗𝑌)

=hhΛ| (𝑋 ⊗𝑌) |𝐼ii +tr(𝐸˜^𝑛(𝑋 ⊗𝑌)), hence

|hhΛ| (𝑋 ⊗𝑌) |𝐼ii| =|tr(𝐸˜^𝑛(𝑋 ⊗𝑌)) | ≤ k𝐸˜k𝐹 · k𝑋 ⊗𝑌k𝐹

≤ 𝜆

𝑛/2

2 k𝑋k𝐹· k𝑌k𝐹 , using (2.4.2). The claim (ii) then follows from (i).

With the following lemma, we prove an upper bound on the overlap of the reduced density matrices𝜌_𝑋and𝜌_𝑌, supported on 2Δ-sites surrounding the boundary, of the global states|Ψ𝑋iand|Ψ𝑌i, respectively.

Lemma 2.5.2. Let𝐴be an MPS tensor of an injective MPS with bond dimension𝐷, and let 𝑋 , 𝑌 ∈ B ( (C^p)^⊗𝑛) be such that the states |Ψ𝑋i = |Ψ(𝐴, 𝑋 , 𝑛)i and |Ψ𝑌i =

|Ψ(𝐴, 𝑌 , 𝑛)i are normalized and orthogonal. LetΔ 𝐷. Let S ={1,2, . . . ,Δ} ∪ {𝑛−Δ+1, 𝑛−Δ+2, . . . , 𝑛} be the subset of2Δspins consisting of Δ systems at the left and andΔsystems at the right boundary. Let 𝜌_𝑋 = tr[𝑛]\S|Ψ𝑋ihΨ𝑋| and 𝜌_𝑌 =tr[𝑛]\S|Ψ𝑌ihΨ𝑌|be the reduced density operators on these subsystems. Then

tr(𝜌_𝑋𝜌_𝑌) ≤ 𝑐𝜆

Δ 2

2

where 𝜆

2 is the second largest eigenvalue of the transfer operator 𝐸 = 𝐸(𝐴) and where𝑐is a constant depending only on the minimal eigenvalue of𝐸 and the bond dimension𝐷.

Figure 2.4: The two expressions in Equation (2.5), where 𝐿, 𝑀, and 𝑅are used to denote the sites defined in (2.5).

Proof. For convenience, let us relabel the systems as (𝐿

1, . . . , 𝐿_Δ) = (1,2, . . . ,Δ) (𝑀

1, . . . , 𝑀_𝑛−

2Δ) = (Δ+1,Δ+2, . . . , 𝑛−Δ) (𝑅

1, . . . , 𝑅_Δ) = (𝑛−Δ+1, 𝑛−Δ+2, . . . , 𝑛)

(2.31) indicating their location on the left, in the middle, and on the right, respectively.

For the tensor productH𝐴 ⊗ H𝐵 of two isomorphic Hilbert spaces, we denote by F^{𝐴 𝐵} ∈ B (H𝐴⊗ H𝐵) the flip-operator which swaps the two systems. The following expressions are visualized in Figure 2.4. We have

tr(𝜌_𝑋𝜌_𝑌) =tr( (𝜌^{𝐿1···𝐿Δ𝑅1···𝑅Δ}

𝑋 ⊗ 𝜌

𝐿⁰

1···𝐿⁰

Δ𝑅⁰

1···𝑅⁰

Δ

𝑌 ) (F^{𝐿 𝐿0} ⊗F^{𝑅 𝑅0})), where F^{𝐿 𝐿0} =F^𝐿₁^𝐿0

1

⊗F^𝐿₂^𝐿0

2

⊗ · · · ⊗F^𝐿Δ𝐿⁰

Δ

, F^{𝑅 𝑅0} =F^𝑅₁^𝑅0

1

⊗F^𝑅₂^𝑅0

2

⊗ · · · ⊗F^𝑅Δ𝑅⁰

Δ

. DefiningF^{𝑀 𝑀0} analogously, 𝐼_{𝑀 𝑀}0 = 𝐼_𝑀

1𝑀⁰

1

⊗ · · · ⊗ 𝐼_𝑀

𝑛−2Δ𝑀⁰

𝑛−2Δ, and similarly 𝐼_{𝐿 𝐿}0

and𝐼_{𝑅 𝑅}0, this can be rewritten (by the definition of the partial trace) as tr(𝜌_𝑋𝜌_𝑌)

=(hΨ^{𝐿 𝑀 𝑅}_𝑋 | ⊗ hΨ_𝑌^{𝐿0𝑀0𝑅0}|) (F^{𝐿 𝐿0} ⊗ 𝐼_{𝑀 𝑀}0 ⊗F^{𝑅 𝑅0}) (|Ψ^{𝐿 𝑀 𝑅}_𝑋 i ⊗ |Ψ_𝑌^{𝐿0𝑀0𝑅0}i)

=(hΨ^{𝐿 𝑀 𝑅}_𝑋 | ⊗ hΨ_𝑌^{𝐿0𝑀0𝑅0}|) (𝐼_{𝐿 𝐿}0 ⊗F^{𝑀 𝑀0} ⊗ 𝐼_{𝑅 𝑅}0) (|Ψ_𝑌^{𝐿 𝑀 𝑅}i ⊗ |Ψ^𝐿_𝑋^0𝑀0𝑅0i).(2.32)

In the last identity, we have used thatF² =𝐼 is the identity.

Reordering and regrouping the systems as (𝐿

1𝐿⁰

1) (𝐿

2𝐿⁰

2) · · · (𝐿_Δ𝐿⁰

Δ) (𝑀

1𝑀⁰

1) (𝑀

2𝑀⁰

2) · · · (𝑀_𝑛−

2Δ𝑀⁰

𝑛−2Δ) (𝑅

1𝑅⁰

1) (𝑅

2𝑅⁰

2) · · · (𝑅_Δ𝑅⁰

Δ),

we observe that|Ψ^{𝐿 𝑀 𝑅}

𝑋 i ⊗ |Ψ^{𝐿0𝑀0𝑅0}

𝑌 iis an MPS with MPS tensor𝐴⊗𝐴and boundary tensor𝑋⊗𝑌and|Ψ^{𝐿 𝑀 𝑅}

𝑌 i ⊗ |Ψ^{𝐿0𝑀0𝑅0}

𝑋 iis an MPS with MPS tensor𝐴⊗𝐴and boundary tensor𝑌 ⊗𝑋. Let us denote the virtual systems of the first MPS by𝑉

1𝑉

2, and those of the second MPS by 𝑊

1𝑊

2, such that the boundary tensors are 𝑋^𝑉1 ⊗𝑌^𝑉2 and 𝑌^𝑊1⊗𝑋^𝑊2, respectively. Let ˆ𝐸 = 𝐸^𝑉1𝑊

1⊗𝐸^𝑉2𝑊

2be the associated transfer operator.

Then we have from (2.5) tr(𝜌_𝑋𝜌_𝑌) =tr

ˆ 𝐸^Δ𝐸ˆ

F^{⊗𝑛−2Δ}𝐸ˆ^Δ h

(𝑋

𝑉1 ⊗𝑌

𝑉2) ⊗ (𝑌^𝑊1 ⊗ 𝑋^𝑊2)i

. (2.33) Recall that𝐸^Δ = |𝐼iihhΛ| ⊕𝐸˜^Δ, where we have

k𝐸˜^Δk𝐹 ≤√︁

𝐷2· k𝐸˜^Δk ≤ 𝐷·𝜆^Δ/2

2 forΔ 𝐷 ,

k |𝐼iihhΛ| k𝐹 =k |𝐼ii k²· k |Λii k² ≤ 𝐷².

In the second line, we use the fact that k |Λii k² = tr(Λ^†Λ) = Í

𝑖𝜆_𝑖² ≤ 1 and k |𝐼ii k² =𝐷². Therefore, we have

𝐸^Δ = ∑︁

𝑏∈{0,1}

𝐻_𝑏, where𝐻

0= |𝐼iihhΛ|and𝐻

1=𝐸˜^Δsatisfy k𝐻

0k𝐹 ≤ 𝐷², and k𝐻

1k𝐹 ≤ 𝐷·𝜆^Δ/2

2 forΔ 𝐷 . (2.34) Note that

ˆ

𝐸^Δ =𝐸^Δ ⊗𝐸^Δ = ∑︁

𝑏1,𝑏

2∈{0,1}

𝐻_𝑏

1 ⊗𝐻_𝑏

2 . Inserting this into (2.5) gives a sum of 16 terms

tr(𝜌_𝑋𝜌_𝑌) ≤ ∑︁

𝑏1,𝑏

2,𝑏

3,𝑏

4∈{0,1}

|𝛼_𝑏

1,𝑏

2,𝑏

3,𝑏

4|, where

𝛼_𝑏

1,𝑏

2,𝑏

3,𝑏

4 =tr

(𝐻^𝑉1𝑊1

𝑏1

⊗ 𝐻^𝑉2𝑊2

𝑏2

)𝐸ˆ

F^{⊗𝑛−2Δ}(𝐻^𝑉1𝑊1

𝑏3

⊗ 𝐻^𝑉2𝑊2

𝑏4

) h (𝑋

𝑉1⊗𝑌

𝑉2) ⊗ (𝑌^𝑊1 ⊗ 𝑋^𝑊2)i .

Consider the term with𝑏_𝑗 =0 for all 𝑗 ∈ {1, . . . ,4}. This is given by 𝛼0,0,0,0=tr

(|𝐼iihhΛ|^𝑉1𝑊1 ⊗ |𝐼iihhΛ|^𝑉2𝑊2)𝐸ˆ

F^{⊗𝑛−2Δ}(|𝐼iihhΛ|^𝑉1𝑊1 ⊗ |𝐼iihhΛ|^𝑉2𝑊2)

· h (𝑋

𝑉1 ⊗𝑌

𝑉2) ⊗ (𝑌^𝑊1 ⊗ 𝑋^𝑊2)i

=hhΛ| (𝑋 ⊗𝑌) |𝐼iihhΛ| (𝑌¯ ⊗ 𝑋) |𝐼ii (hhΛ| ⊗ hhΛ|)𝐸ˆ

F^{⊗𝑛−2Δ}(|𝐼ii ⊗ |𝐼ii).(2.35) By inserting this into (2.5), we get with Lemma 2.5.1 (ii) and the Cauchy-Schwarz inequality

|𝛼

0,0,0,0|=𝑂(𝜆^𝑛

2) ·

hhΛ| ⊗ hhΛ|)𝐸ˆ

F^{⊗𝑛−2Δ}(|𝐼ii ⊗ |𝐼ii

=𝑂(𝜆^𝑛

2) · k |Λii ⊗ |Λii k · k𝐸ˆ

F^{⊗𝑛−2Δ}(|𝐼ii ⊗ |𝐼ii k . With Lemma 2.4.4, this can further be bounded as

|𝛼

0,0,0,0| =𝑂(𝜆^𝑛

2) · k |Λii k²· k |𝐼ii k²· kF^{⊗𝑛−2Δ}k · k𝐸^𝑛−2Δk .

SincekFk =1 andk |Λii k =𝑂(1), k |𝐼ii k =𝑂(1) andk𝐸^𝑛−2Δk =𝑂(1)(cf. (2.4.3)), we conclude that

|𝛼

0,0,0,0|=𝑂(𝜆^𝑛

2) . (2.36)

The remaining terms|𝛼_𝑏

1,𝑏

2,𝑏

3,𝑏

4|with(𝑏

1, 𝑏

2, 𝑏

3, 𝑏

4) ≠ (0,0,0,0)can be bounded as follows: using inequality (2.5), we have

|𝛼_𝑏

1,𝑏

2,𝑏

3,𝑏

4|= tr

(𝐻_𝑏

1 ⊗ 𝐻_𝑏

2)𝐸ˆ

F^{⊗𝑁−2Δ}(𝐻_𝑏

3 ⊗𝐻_𝑏

4)h

(𝑋 ⊗𝑌) ⊗ (𝑌 ⊗ 𝑋)i

≤ k𝐻_𝑏

1 ⊗ 𝐻_𝑏

2k𝐹 · k𝐸

F^{⊗𝑁−2𝑛}k𝐹 · k𝐻_𝑏

3 ⊗ 𝐻_𝑏

4k𝐹 · k𝑋 ⊗𝑌 ⊗𝑌 ⊗ 𝑋k𝐹

=k𝑋k²_𝐹 · k𝑌k²_𝐹 ·©

­

« Ö4

𝑗=1

k𝐻_𝑏

𝑗k𝐹ª

®

¬

· k𝐸

F^{⊗𝑛−2Δ}k𝐹

=𝑂(𝜆^Δ/2

2 ) · k𝑋k²_𝐹· k𝑌k²_𝐹 · k𝐸

F^{⊗𝑛−2Δ}k𝐹 , where we use (2.5) and the assumption that (𝑏

1, 𝑏

2, 𝑏

3, 𝑏

4) ≠ (0,0,0,0). We use Lemma 2.4.4 and (2.4.3) to get the upper bound k𝐸

F^{⊗𝑛−2Δ}k ≤ 𝐷²k𝐹^{⊗𝑛−2Δ}k · k𝐸^𝑛−𝐷k =𝑂(1). Thus

|𝛼_𝑏

1,𝑏

2,𝑏

3,𝑏

4| =𝑂(𝜆^Δ/2

2 ) for (𝑏

1, 𝑏

2, 𝑏

3, 𝑏

4) ≠ (0,0,0,0) . (2.37) Combining (2.5) with (2.5), we conclude that

|tr(𝜌_𝑋𝜌_𝑌) | ≤ ∑︁

𝑏1,𝑏

2,𝑏

3,𝑏

4∈{0,1}

|𝛼_𝑏

1,𝑏

2,𝑏

3,𝑏

4| ≤ |𝛼

0,0,0,0| +15 max

(𝑏₁,𝑏

2,𝑏

3,𝑏

4)≠(0,0,0,0)

|𝛼_𝑏

1,𝑏

2,𝑏

3,𝑏

4|

=𝑂(𝜆^Δ/2

2 ) .

The claim follows from this.

Recall that we call (a family of subspaces) C ⊂ (C^p)^⊗𝑛 an approximate error- detection code if it is an (𝜖 , 𝛿) [ [𝑛, 𝑘 , 𝑑]]-code with𝜖 → 0 and𝛿 → 0 for𝑛 → ∞. Our main result is the following:

Theorem 2.5.3. LetC ⊂ (C^p)^⊗𝑛be an approximate error-detecting code generated from a translation-invariant injective MPS of constant bond dimension𝐷by varying boundary conditions. Then the distance ofCis constant.

Proof. LetC =C𝑛 ⊂ (C^p)^⊗𝑛 be a (family of) subspace(s) of dimension^p𝑘 defined by an MPS tensor 𝐴by choosing different boundary conditions, i.e.,

C𝑛={|Ψ(𝐴, 𝑋 , 𝑛)i | 𝑋 ∈ X} ⊂ (C^p)^⊗𝑛

for some (fixed) subspaceX ⊂ B (C^𝐷). For the sake of contradiction, assume that C𝑛is an (𝜖_𝑛, 𝛿_𝑛) [ [𝑛, 𝑘 , 𝑑_𝑛]]-code with

𝜖_𝑛, 𝛿_𝑛 →0 and code distance𝑑_𝑛→ ∞ for𝑛→ ∞. (2.38) Let |Ψ𝑋i = |Ψ(𝐴, 𝑋 , 𝑛)i,|Ψ𝑌i = |Ψ(𝐴, 𝑌 , 𝑛)i ∈ C be two orthonormal states defined by choosing different boundary conditions 𝑋 , 𝑌 ∈ X. From Lemma 2.5.2, we may chooseΔsufficiently large such that the reduced density operators 𝜌_𝑋, 𝜌_𝑌 on𝑑sites surrounding the boundary satisfies

tr(𝜌_𝑋𝜌_𝑌) ≤ 𝑐𝜆^𝑑/4

2 for all 𝑑 ≥2Δ. (2.39)

We note thatΔonly depends on the transfer operator and is independent of𝑛. Fix any constant𝜖 , 𝛿 ∈ (0,1)and choose some𝑑 ≥ 2Δsufficiently large such that

𝜁(𝜌_𝑋, 𝜌_𝑌) :=𝑐 𝐷²𝜆^𝑑/4

2

satisfies

𝜖 < 1−10𝜁 and 𝛿 < (1−𝜁)². (2.40) Since by assumption𝑑_𝑛 → ∞, there exists some𝑁

0 ∈Nsuch that 𝑑_𝑛 > 𝑑 for all𝑛 ≥ 𝑁

0. (2.41)

Combining (2.5), (2.5), and (2.5) with Lemma 2.3.6, we conclude thatC𝑛is not an (𝜖 , 𝛿) [ [𝑛, 𝑘 , 𝑑_𝑛]]-code for any𝑛 ≥ 𝑁

0.

By assumption (2.5), there exists some𝑁

1 ∈Nsuch that 𝜖_𝑛 < 𝜖 and 𝛿_𝑛 < 𝛿 for all𝑛 ≥ 𝑁

1. Let us set𝑁 =max{𝑁

0, 𝑁

1}. Then we obtain thatC𝑛is not an (𝜖_𝑛, 𝛿_𝑛) [ [𝑛, 𝑘 , 𝑑_𝑛]]- code for any𝑛 ≥ 𝑁, a contradiction.

In terms of the TQO-1 condition (cf. [57]), Theorem 2.5.3 shows the absence of topological order in 1D gapped systems. The theorem also tells us that we should not restrict our attention to the ground space of a local Hamiltonian when looking for quantum error-detecting codes.⁵ In the following sections, we bypass this no-go result by extending our search for codes to low-energy states. In particular, we show that single quasi-particle momentum eigenstates of local gapped Hamiltonians and multi-particle excitations of the gapless Heisenberg model constitute error-detecting codes. See Sections 2.6 and 2.7, respectively.