Transfer Matrix Techniques

QUANTUM ERROR-DETECTION AT LOW ENERGIES

2.4 On Expectation Values of Local Operators in MPS

2.4.2 Transfer Matrix Techniques

Here we establish some essential statements for the analysis of transfer operators. In Section 2.4.2, we introduce generalized (non-standard) transfer operators: these can be used to express the matrix elements of the form hΨ|𝐹|Ψ⁰i of local operators 𝐹 with respect to pairs of MPS (Ψ,Ψ⁰). In Section 2.4.2, we establish bounds on the norm of such operators. Relevant quantities appearing in these bounds are the second largest eigenvalue𝜆

2of the transfer matrix, as well as the sizes of its Jordan blocks.

More General and Mixed Transfer Operators

Consider a single-site operator 𝑍 ∈ B (C^p). The generalized transfer matrix𝐸_𝑍 ∈ B (C^𝐷 ⊗C^𝐷)is defined as

𝐸_𝑍 =∑︁

𝑛,𝑚

h𝑚|𝑍|𝑛i𝐴_𝑚 ⊗ 𝐴_𝑛 . (2.17) We further generalize this as follows: if 𝑍

1, . . . , 𝑍_𝑑 ∈ B (C^p), then 𝐸_𝑍

1⊗···⊗𝑍𝑑 ∈ B (C^𝐷 ⊗C^𝐷)is the operator

𝐸_𝑍

1⊗···⊗𝑍𝑑 =𝐸_𝑍

1· · ·𝐸_𝑍

𝑑 .

This definition extends by linearity to any operator 𝐹 ∈ B ( (C^p)^⊗𝑑), and gives a corresponding operator𝐸_𝐹 ∈ B (C^𝐷 ⊗C^𝐷). The tensor network diagrams for these definitions are given in Figure 2.2, and the composition of the corresponding maps is illustrated in Figure 2.3.

In the following, we are interested in inner productshΨ(𝐴, 𝑋 , 𝑛) |Ψ(𝐵, 𝑌 , 𝑛)i of two MPS, defined by local tensors 𝐴 and 𝐵, with boundary matrices 𝑋 and𝑌, which may have different bond dimensions 𝐷

1and 𝐷

2, respectively. To analyze these, it is convenient to introduce an “overlap” transfer operator 𝐸 = 𝐸(𝐴, 𝐵) which now depends on both MPS tensors 𝐴and𝐵. First, we define𝐸 ∈ B (C^𝐷1 ⊗C^𝐷2) by

𝐸 =

p

∑︁

𝑚=1

𝐴_𝑚 ⊗ 𝐵_𝑚 .

Figure 2.3: The product 𝐸_𝑍

1⊗𝑍

2 = 𝐸_𝑍

1𝐸_𝑍

2 of two transfer operators. Left- multiplication by an operator corresponds to attaching the corresponding diagram on the left.

The definition of 𝐸_𝑍 for 𝑍 ∈ B (C^p) is analogous to Equation (2.4.2), but with appropriate substitutions. We set

𝐸_𝑍 =∑︁

𝑛,𝑚

h𝑚|𝑍|𝑛i𝐴_𝑚 ⊗ 𝐵_𝑛 .

Starting from this definition, the expression𝐸_𝐹 ∈ B (C^𝐷1⊗C^𝐷2)for𝐹 ∈ B ( (C^p)^⊗𝑑) is then defined analogously as before.

Norm Bounds on Generalized Transfer Operators

A first key observation is that the (operator) norm of powers of any transfer operator scales (at most) as a polynomial in the number of physical spins, with the degree of the polynomial determined by the size of the largest Jordan block. We need these bounds explicitly and start with the following simple bounds.

Below, we often consider families of parameters depending on the system size 𝑛, i.e., the total number of spins. We write𝑚 ℎas a shorthand for a parameter𝑚

“being sufficiently large” compared to another parameter ℎ. More precisely, this signifies that we assume that |ℎ/𝑚| → 0 for𝑛 → ∞, and that by a corresponding choice of a sufficiently large𝑛, the term |ℎ/𝑚|can be made sufficiently small for a given bound to hold. Oftentimesℎwill in fact be constant, with𝑚 → ∞as𝑛→ ∞. Lemma 2.4.1. For 𝑚 > ℎ, the Frobenius norm of the 𝑚-th power (𝜆 𝐼+𝑁)^𝑚 of a Jordan block𝜆 𝐼 +𝑁 ∈ B (C^ℎ) with eigenvalue𝜆, such that |𝜆| ≤ 1, and size ℎ is bounded by

k (𝜆 𝐼+𝑁)^𝑚k_𝐹 ≤ 3ℎ^3/2𝑚^ℎ−1|𝜆|^{𝑚−(ℎ−1)} . (2.18) Furthermore,

k (𝜆 𝐼+𝑁)^𝑚 k ≤4𝑚^ℎ−1 for𝑚 ℎ . (2.19)

Proof. Forℎ =0 the claim is trivial. Assume thatℎ >1. Because 𝑁^ℎ =0 and𝑁^𝑟 has exactlyℎ−𝑟 non-zero entries for𝑟 < ℎ, we have

k (𝜆 𝐼_ℎ+𝑁)^𝑚k_𝐹 ≤

ℎ−1

∑︁

𝑟=0

𝑚 𝑟

|𝜆|^𝑚−𝑟 k𝑁^𝑟k_𝐹

≤ |𝜆|^𝑚· |𝜆|^−(ℎ−1)

ℎ−1

∑︁

𝑟=0

𝑚 𝑟

(ℎ−𝑟)^1/2

≤ ℎ^1/2|𝜆|^𝑚 · |𝜆|^−(ℎ−1)

ℎ−1

∑︁

𝑟=0

𝑚 𝑟

.

Since the right hand side is maximal for𝑟 =ℎ−1, and the binomial coefficient can be bounded from above by ^𝑚𝑟

≤ ^𝑒𝑚

𝑟

^𝑟

≤ 3·𝑚^𝑟, we obtain

ℎ−1

∑︁

𝑟=0

𝑚 𝑟

≤ 3ℎ·𝑚^ℎ−1, hence, the first claim follows.

For the second claim, recall that the entries of the𝑚-th power of a Jordan block are ( (𝜆 𝐼_ℎ+𝑁)^𝑚)_𝑝,𝑞 =











𝑚 𝑞−𝑝

𝜆^{𝑚+(𝑝−𝑞)} if𝑞 ≥ 𝑝

0 otherwise

(2.20)

for𝑝, 𝑞 ∈ [ℎ]. This means that if|𝜆|=1, the maximum matrix element| ( (𝜆 𝐼_ℎ+𝑁)^𝑚)_𝑝,𝑞|=

𝑚 ℎ−1

is attained for (𝑝, 𝑞) = (1, ℎ). Using the Cauchy-Schwarz inequality, it is straightforward to check that

k (𝜆 𝐼+𝑁)^𝑚k ≤ ℎmax

𝑝,𝑞

| (𝜆 𝐼+𝑁)^𝑚_𝑝,𝑞| =ℎ· 𝑚

ℎ−1

= ℎ (ℎ−1)!

𝑚! (𝑚− (ℎ−1))!

. Since _(ℎ−^ℎ_1)! ≤ 2 forℎ ∈Nand

𝑚! (𝑚− (ℎ−1))!

=𝑚^ℎ−1(1+𝑂(ℎ/𝑚)) ≤2𝑚^ℎ−1 for𝑚 ℎ , the claim follows.

Now, we apply Lemma 2.4.1 to (standard and mixed) transfer operators. It is convenient to state these bounds as follows. The first two statements are about the scaling of the norms of powers of 𝐸; the last statement is about the magnitude of matrix elements in powers of𝐸.

Lemma 2.4.2. Let𝜌(𝐸)denote the spectral radius of a matrix𝐸 ∈ B (C^𝐷1⊗C^𝐷2). (i) Suppose𝜌(𝐸) ≤ 1. Letℎ^∗be the size of the largest Jordan block(s) of𝐸. Then

k𝐸^𝑚k ≤ 4𝑚^ℎ−1 for𝑚 ℎ . (ii) If𝜌(𝐸) <1, then

k𝐸^𝑚k𝐹 ≤ 𝜌(𝐸)^𝑚/2 for𝑚 𝐷

1𝐷

2. We will often usek𝐸^𝑚k𝐹 ≤1as a coarse bound.

(iii) Suppose that 𝜌(𝐸) =1. Let ℎ^∗ denote the size of the largest Jordan block(s) in 𝐸. For 𝑝, 𝑞 ∈ [𝐷

1𝐷

2], let (𝐸^𝑚)𝑝,𝑞 denote the matrix element of 𝐸^𝑚 with respect to the standard computational basis {|𝑝i}^𝐷_𝑝=^1𝐷2

1 . Then the following holds: for all 𝑝, 𝑞 ∈ [𝐷

1𝐷

2], there is a constant 𝑐_𝑝,𝑞 with 𝑐_𝑝,𝑞 = 𝑂(1) as 𝑚→ ∞and someℓ ∈ {1, . . . , ℎ^∗}such that

| (𝐸^𝑚)𝑝,𝑞|=𝑐𝑚^ℓ−1(1+𝑂(𝑚⁻¹)) .

Proof. For𝜆∈spec(𝐸), let us denote by𝜆 𝐼_ℎ(𝜆)+𝑁_ℎ(𝜆)the associated Jordan block, whereℎ(𝜆)is its size. Then

k𝐸^𝑚k = max

𝜆∈spec(𝐸)

k (𝜆 𝐼_ℎ(𝜆)+𝑁_ℎ(𝜆))^𝑚k ≤ max

𝜆∈spec(𝐸)

k (𝜆 𝐼_ℎ(𝜆) +𝑁_ℎ(𝜆))^𝑚k𝐹

where we assumed that𝑚 ℎ^∗ ≥ ℎ(𝜆), |𝜆| ≤ 1 and (2.4.1). This shows claim (i).

Claim (ii) immediately follows from (2.4.1) and the observation that𝑚^𝐷1𝐷

2−1𝜌(𝐸)^𝑚 = 𝑂(𝜌(𝐸)^𝑚/2).

For the proof of statement (iii), observe that matrix elements (2.4.2) of a Jordan block (matrix) with eigenvalue𝜆(with|𝜆|=1) of sizeℎscale as

( (𝜆 𝐼+𝑁)^𝑚)𝑝,𝑞

= 1 (𝑞− 𝑝)!

𝑚! (𝑚− (𝑞−𝑝))!

= 1 (𝑞− 𝑝)!

𝑚^𝑞−𝑝(1+𝑂(1/𝑚)) for𝑞 > 𝑝and are constant otherwise. Because𝑞− 𝑝 ∈ {1, . . . , ℎ−1}when𝑞 > 𝑝, this is of the form 𝑚^ℓ(1+𝑂(1/𝑚)) for some ℓ ∈ [ℎ− 1]. Since 𝐸^𝑚 is similar (as a matrix) to a direct sum of such powers of Jordan blocks, and the form of this scaling does not change under linear combination of matrix coefficients, the claim follows.

Now let us consider the case where 𝐸 = |ℓiihh𝑟| ⊕𝐸˜ is the transfer operator of an injective MPS, normalized with maximum eigenvalue 1. Let 𝜆

2 < 1 denote the second largest eigenvalue. Without loss of generality, we can take the MPS to be in canonical form, so that𝐸 has a unique right fixed-point given by the identity matrix 𝐼 and a unique left fixed-point given by some positive-definite diagonal matrixΛ with unit trace. We can then write the Jordan decomposition of the transfer matrix as

𝐸 =|𝐼iihhΛ| ⊕𝐸 ,˜ (2.21)

where |𝐼ii and |Λii denotes the vectorization of 𝐼 and Λ, respectively, and where

˜

𝐸 denotes the remaining Jordan blocks of 𝐸. Note that powers of 𝐸 can then be expressed as

𝐸^𝑚 =|𝐼iihhΛ| ⊕𝐸˜^𝑚 .

We can bound the Frobenius norm of the transfer matrix as

k𝐸^𝑚k²_𝐹 =k |𝐼iihhΛ| ⊕ 𝐸˜^𝑚k²_𝐹 =tr(𝐼)tr(Λ²) + k𝐸˜^𝑚k²_𝐹 ≤ 𝐷+ k𝐸˜^𝑚k²_𝐹, where tr(𝐼) =𝐷and tr(Λ²) ≤tr(Λ)²=1. In particular, since𝜌(𝐸˜) =𝜆

2, we obtain from Lemma 2.4.2(ii) that

k𝐸˜^𝑚k𝐹 ≤𝜆^𝑚/2

2 for𝑚 𝐷 . (2.22)

This implies the following statement:

Lemma 2.4.3. The transfer operator𝐸 of an injective MPS satisfies k𝐸^𝑚k𝐹 ≤

√

𝐷+1 for𝑚 𝐷 . (2.23)

We also need a bound on the norm k𝐸^†

𝐹(𝜓

1⊗ 𝜓

2) k, where 𝐸 is a mixed transfer operator,𝐹 ∈ B ( (C^p)^⊗𝑑) is an operator acting on𝑑 sites, and where𝜓_𝑗 ∈C^𝐷𝑗 for

𝑗 =1,2.

Lemma 2.4.4. Let𝐸

1= 𝐸(𝐴)and𝐸

2=𝐸(𝐵) be the transfer operators associated with the tensors 𝐴 and 𝐵, respectively, with bond dimensions 𝐷

1 and 𝐷

2. Let 𝐸 = 𝐸(𝐴, 𝐵) ∈ B (C^𝐷1 ⊗ C^𝐷2) be the combined transfer operator. Let 𝜓

1 ∈ C^𝐷1 and𝜓

2∈C^𝐷2 be unit vectors. Then k (𝐸_𝐹)^†(𝜓

1⊗𝜓

2) k ≤ k𝐹k

√︃

k𝐸^𝑑

1k · k𝐸^𝑑

2k , (2.24)

k𝐸_𝐹(𝜓

1⊗𝜓

2) k ≤ k𝐹k

√︃

k𝐸^𝑑

1k · k𝐸^𝑑

2k , (2.25)

k𝐸_𝐹k𝐹 ≤ 𝐷

1𝐷

2k𝐹k

√︃

k𝐸^𝑑

1k · k𝐸^𝑑

2k , (2.26)

for all𝐹 ∈ B ( (C^p)^⊗𝑑).

Proof. Writing matrix elements in the computational basis as 𝐹_𝑗

1···𝑗𝑑,𝑖

1···𝑖𝑑 :=h𝑗

1· · ·𝑗_𝑑|𝐹|𝑖

1· · ·𝑖_𝑑i, we have

𝐸_𝐹 = ∑︁

(𝑖

1,...,𝑖_𝑑),(𝑗

1,..., 𝑗_𝑑)

𝐹_𝑗

1···𝑗_𝑑,𝑖

1···𝑖_𝑑(𝐴_𝑗

1 ⊗ 𝐵_𝑗

1) (𝐴_𝑗

2 ⊗𝐵_𝑗

2) · · · (𝐴_𝑗

𝑑 ⊗ 𝐵_𝑗

𝑑). Therefore,

(𝐸_𝐹)^† = ∑︁

(𝑖

1,...,𝑖_𝑑),(𝑗

1,..., 𝑗_𝑑)

𝐹_𝑗

1···𝑗𝑑,𝑖

1···𝑖𝑑(𝐴^†

𝑗𝑑

⊗𝐵^†

𝑗𝑑

) · · · (𝐴^†

𝑗2

⊗ 𝐵^†

𝑗2

) (𝐴^†_𝑗

1 ⊗𝐵^†

𝑗1

)

= ∑︁

(𝑖

1,...,𝑖_𝑑),(𝑗

1,..., 𝑗_𝑑)

(𝜋 𝐹 𝜋^†)𝑗𝑑···𝑗

1,𝑖𝑑···𝑖₁(𝐴^†

𝑗𝑑

⊗ 𝐵^†

𝑗𝑑

) · · · (𝐴^†

𝑗2

⊗𝐵^†

𝑗2

) (𝐴^†

𝑗1

⊗ 𝐵^†

𝑗1

) ,

where𝜋is the permutation which maps the 𝑗-th factor in the tensor product(C^p)^⊗𝑛 to the (𝑛− 𝑗 +1)-th factor, and where 𝐹 is obtained by complex conjugating the matrix elements in the computational basis. This means that

(𝐸_𝐹)^†= 𝐸^†

𝜋 𝐹 𝜋^†

, (2.27)

with 𝐸^† being the mixed transfer operator 𝐸^† = 𝐸(𝐴^†, 𝐵^†) obtained by replacing each𝐴_𝑗 (respectively𝐵_𝑗) with its adjoint.

Now consider

k (𝐸_𝐹)^†(𝜓

1⊗𝜓

2) k²= (h𝜓

1| ⊗ h𝜓

2|)𝐸_𝐹(𝐸_𝐹)^†(|𝜓

1i ⊗ |𝜓

2i)

= (h𝜓

1| ⊗ h𝜓

2|)𝐸_𝐹𝐸^†

𝜋 𝐹 𝜋^†

(|𝜓

1i ⊗ |𝜓

2i) . This can be represented diagrammatically as

k (𝐸_𝐹)^†(𝜓

1⊗𝜓

2) k²= .

In particular, we have k (𝐸_𝐹)^†(𝜓

1⊗𝜓

2) k² = h𝜒| (𝐹 ⊗𝐼^⊗𝑑) (𝐼^⊗𝑑 ⊗𝜋 𝐹 𝜋^†) |𝜑i, (2.28)

where𝜑, 𝜒∈ (C^p)^⊗2𝑑 are defined as

|𝜙i= ,

|𝜒i= .

It is straightforward to check that k𝜒k²= (h𝜓

1| ⊗ h𝜓

1|)𝐸^𝑑

1(𝐸^†

1)^𝑑(|𝜓

1i ⊗ |𝜓

1i) , k𝜑k²= (h𝜓

2| ⊗ h𝜓

2|)𝐸^𝑑

2(𝐸^†

2)^𝑑(|𝜓

2i ⊗ |𝜓

2i) . Since k (𝐸^†

𝑗)^𝑑k = k𝐸^𝑑

𝑗k for 𝑗 = 1,2, it follows with the submultiplicativity of the operator norm that

k𝜒k² ≤ k𝐸^𝑑

1k², k𝜑k² ≤ k𝐸^𝑑

2k². (2.29)

Applying the Cauchy-Schwarz inequality to (2.4.2) yields k (𝐸_𝐹)^†(𝜓

1⊗𝜓

2) k²≤ k (𝐹^†⊗ 𝐼^⊗𝑑)𝜒k · k (𝐼^⊗𝑑 ⊗𝜋 𝐹 𝜋^†)𝜑k

≤ k𝐹k²· k𝜒k · k𝜑k,

where we used the fact that the operator norm satisfies k𝐹^†k = k𝐹k = k𝐹k and k𝐼 ⊗ 𝐴k =k𝐴k. The claim (2.4.4) follows from this and (2.4.2).

The claim (2.4.4) follows analogously by using Equation (2.4.2). Finally, the claim (2.4.4) follows from (2.4.4) and

k𝐸_𝐹k² = ∑︁

𝛼1,𝛼

2∈[𝐷

1]

∑︁

𝛽1, 𝛽

2∈[𝐷

2]

| (h𝛼

1| ⊗ h𝛽

1|)𝐸_𝐹(|𝛼

2i ⊗ |𝛽

2i) |²

≤ ∑︁

𝛼1,𝛼

2∈[𝐷₁]

∑︁

𝛽1, 𝛽

2∈[𝐷

2]

k𝐸_𝐹(|𝛼

2i ⊗ |𝛽

2i) k²

≤ 𝐷²

1𝐷²

2max

𝛼, 𝛽

k𝐸_𝐹(|𝛼

2i ⊗ |𝛽

2i) k², where we employed the orthonormal basis{|𝛼i}𝛼∈[𝐷

1] and{|𝛽i}𝛽∈[𝐷

2] forC^𝐷1and C^𝐷2, respectively, and applied the Cauchy-Schwarz inequality.

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