Quasi-exact quantum computation

We study quasi-exact quantum error correcting codes and quantum computation with them. Here `quasi-exact' means it becomes the same as the exact cases under proper limits. We find that the tradeoff between universality and transversality of the set of quantum gates, due to Eastin and Knill, does not persist in the quasi-exact scenario. A class of covariant quasi-exact codes is defined which proves to support universal and transversal set of gates. This work opens the possibility of quasi-exact universal quantum computation with universality, transversality, and fault tolerance.


Introduction.
Fault-tolerant quantum computation [1][2][3][4] requires quantum codes with nontrivial errorcorrection features.It proves to be difficult for a single quantum code to support universal quantum computation, i.e., a universal set of quantum gates.This is stated as a tradeoff between universality and transversality, i.e., gates of product form U = ⊗ j U j , first proved for stabilizer codes [5,6], and extended for general exact quantum codes [7].Such a tradeoff is further explored for topological stabilizer codes with local stabilizers [8,9], in which setting transversal gates are extended to gates described by finite-depth local unitary circuits [10][11][12].These tradeoffs are often known as 'no-go' theorems.
In this work, we explore methods to circumvent the no-go theorems.Methods in literature mostly focus on relaxing transversality.A common method is to use faulttolerant measurements, such as magic-state injection [13] and code switching [14].Another method is to use concatenation [15,16], which defines different transversality for the concatenated codes.Quantum computing by non-abelian anyons [17], realized in lattice systems, implement braidings by finite-depth local circuits [18][19][20][21][22], hence not transversal.
Instead of transversality, in this work we relax the requirement on universality.Universality means that the group SU (d) can be realized efficiently to arbitrary accuracy based on a discrete universal gate set.The Solovay-Kitaev algorithm [23] ensures that arbitrary accuracy can be achieved efficiently.We realize that such a definition of universality is strong since it allows arbitrarily long accurate quantum computation.However, in practice a computation is usually not arbitrarily long.As a result, we introduce a weak version, coined 'quasi-exact' universality which normally achieves a fixed accuracy but can be made arbitrarily exact under a certain limit.To achieve this, we find that quasi-exact universality can be supported on quasi-exact quantum codes.
There are precursors for the quasi-exact setting we consider.There are various approximate quantum codes [24,25] designed for different tasks.The approximation may rely on limited error models or cannot approach the exact case.Conditions for approximate quantum error correction (QEC) have been studied [26][27][28][29].However, the degree of approximation may not be well bounded, hence not useful for accurate quantum computation.The quasiexact scenario can be viewed as a restricted version of approximate cases so that it can approach the exact case for accurate quantum computation, i.e., the accuracy needs to be bounded or controllable.
In this work we find a nontrivial class of quasi-exact codes which support a set of universal and transversal gates.This class of codes are from SU (d) generalizations of valence-bond solid (VBS) models [30][31][32][33], which appear as ground states of 1D local Hamiltonians with symmetry-protected topological (SPT) orders [34][35][36][37][38] and have proven to be useful in the measurementbased model [39][40][41][42].This class of codes are Lie-group covariant codes, the power of which have been demonstrated recently [43][44][45][46][47].The SPT order enables covariance and quasi-exact universality, and the local errors we consider can be erasure or arbitrary ones not constrained by the symmetry.The Eastin-Knill no-go theorem [7] applies to the quasi-exact setting, except that the number of distinct logical gates can increase to be arbitrarily large leading to the realization of universality, contrary to the exact setting for which the size of the logical gate set is fixed.Our work demonstrates the possibility of faulttolerant quantum computation with (quasi-)universality and transversality.
Quasi-exact codes.A quantum code can be defined by an encoding operator described by an isometry V : H L → H from the logical space H L to the physical space H such that V † V = 1 and V V † = P , for P as the projector onto the code space C ⊂ H.The code dimension is denoted as d L := dimH L = dimC, and the physical dimension is QEC refers to the correction of a set of error operators {E i } on H so that errors cannot induce or disturb any nontrivial logical operations on C. Given {E i }, the correctability condition is [48] Meanwhile, the condition for detection is P E i P = e i P ∝ P, e i ∈ C. Each error operator E i could be local or nonlocal.When {E i } is a local basis of the space of bounded operators on a local site B(H n ) (the subscript n is omitted for error operators E i ) for any n, then for any F j ∈ B(H n ), condition (1) implies P F † j F l P ∝ P , which means arbitrary local error can be corrected.
A given error set {E i } forms a quantum channel, so the error correction finds its inverse on the code space.The condition (1) has been extended to the case when a given channel can only be corrected approximately with accuracy ǫ [26][27][28][29] (see Appendix 1).To define good approximate quantum codes, we need additional structures so that the accuracy ǫ is always bounded and small enough.Here we allow two sources of approximation.First, we allow approximate isometry for the encoding operator such that it becomes exact in a certain limit.Second, we allow approximate error correction for a set of errors such that it becomes exact in a certain limit.We do not restrict on the forms of limits which would depend on the specific codes, and the two limits above could be different in general.We term the cases as quasiexact encoding and quasi-exact QEC.We will use 'quasi' as a shorthand for 'quasi-exact', and use 'quasi code' as a shorthand for a code with either quasi encoding or quasi QEC, or both.
Definition 1 (Quasi encoding).A quasi encoding is defined by a quasi isometry V : H L → H with V † V = 1+∆, and δ-isometric for 0 ≤ d(V † V, 1) ≤ δ, and it becomes exact when δ → 0 in a certain limit, for a proper distance measure d.
The quasi projector onto the code space C ⊂ H is P = V V † and P 2 = P + K, K := V(∆) = V ∆V † , and it becomes an exact projector when δ → 0. The matrix ∆ is of dimension d L and can be specified by V .Definition 2 (Quasi correction).A set of errors {E i }, described as a channel N , is ǫ-correctable when there exists a recovery channel R such that d(V † RN V, 1) ≤ ǫ, and quasi correctable when ǫ → 0 in a certain limit.
Note the distance measure d here shall be the same as, or equivalent to the measure for δ above.Theorem 1.A set of errors {E i } on a code space P is quasi correctable iff for [a ij ] as a quantum state and P B ij P → 0 in a certain limit.
Proof.Sufficiency.Given the condition we need to find a recovery channel R. Similar with the exact case [48], we first use a unitary matrix to diagonalize [a ij ], and then with d kl := d k ≥ 0 for k = l and d kl = 0 for k = l.For , and R as the set {R k }.For any logical state σ C , we find For σ C = P σP , k d k = 1, P BP = 0 with B := i B ii and with (3), we find This means the logical state is recovered up to the quasi projector P and a perturbation described by a map P B kl P }, which does not need to be trace preserving.We need to measure the distance between RN and identity channel.To this end, we map back to the logical space H L by V † .With σ C := V σV † = V(σ), we find for Bkl : We see that the infidelity depends on the size of tr Bkl and the cardinality of the set {k}, i.e., the size of the environment, dimE.Let β := max kl tr Bkl .If the system size is N and local site dimension is d q , then dimE = poly(N, d q ), i.e., a polynomial of N and d q .As a result, If β decays with either N or d q , or both of them, or some other parameters, denoted as λ, faster than the increase with N or d q , then in the large-N , or large-d q , or large-λ limit, the entanglement fidelity becomes 1.
Necessity.Suppose there exists a recovery channel R such that Then the set {R k F l P } is the same representation as the set {P, B i P } for a channel.Similar with the exact case [48], there exists an isometry that maps between the two sets.From easy algebra the condition (2) is obtained.
As β is always small or bounded and tends to zero in a certain limit, the quasi correction of {E i } also implies quasi correction of other operators in its algebra.We see that the quasi-exact scenario is a slight perturbation of the exact case, and similarly, the recovery channel R can be completed to be trace preserving.In addition, the definition of code distance and other relevant quantities have to be modified accordingly.A quasi code distance d C = 2t + 1 can be defined if it can quasi correct up to t errors.Examples of quasi codes can be found in Appendix 2.
Quasi universality.Now we introduce the notion of quasi universality, which is suitable for computing with quasi codes, and can also be employed in other settings.Universality means that on a logical space of dimension d, the whole group SU (d) can be realized [49].The Solovay-Kitaev algorithm [23] proves that, given any U ∈ SU (d), there exists an efficient classical algorithm that produces an efficient sequence of gates U ′ = i U i with the distance d(U, U ′ ) ≤ ǫ for arbitrarily small accuracy ǫ.
In the setting of universality above, the logical identity gate 1 is not explicitly written but implied.However, there is no perfect 1 for quasi codes due to the QEC features (see Eq. ( 6)).The leftover from QEC is random, which means that the logical identity gate 1 is replaced by a set of gates that are close to it, with a proper distance measure such as entanglement fidelity, trace distance, or operator norm.Denote this set as I η for η as a measure of distance from 1.The set I η is an equivalence class of logical gates, and any gate 1 η ∈ I η is treated the same as 1.We call such a set a 'gate-cell', or just 'cell', and η as the 'size' or accuracy of it.Clearly it also induces gate-cells for nontrivial logical gates.
Therefore, the whole group SU (d) can be partitioned into a collection of non-overlapping gate-cells of various sizes; denote this set as SU (d) η , and it is clear that SU (d) η ⊂ SU (d).The partition into cells can be nonuniquely chosen and shall be kept fixed in order to define distinct logical gates, as we will see later.
Definition 4 (Quasi universality).A computation on a quasi code of dimension d and accuracy η is quasi universal iff the whole group SU (d) can be realized when η vanishes.
Suppose we need to approximate U ∈ SU (d) to accuracy ̟.What would be the relation between ̟ and η?For quasi codes that any gate U can be realized directly, ̟ shall be in the same order of η.For the setting that each gate U is built from a product of gates from a universal gate set, denoted as S, ̟ is lower bounded by η since the leftover from QEC can accumulate.The accuracy for different gates could be different.For instance, with This means that arbitrarily large computation cannot be reliably done with quasi codes.In general, given any U ∈ SU (d), first use algorithms [23] to approximate U by U ′ = L J U LJ , for U LJ as gates from a universal gate set S, L as the label of layers relating to the time of computation, and J as the label of gates in each layer relating to the size of computation.Denote the distance d(U, U ′ ) as ̟ 0 , which could be as small as zero.As each U LJ will be approximated by U LJ,η , we see that As a result, we could obtain a lower bound for the accuracy which shall be as expected.For instance, for a quasi code that η is exponentially small with its system size N , the lower bound on ̟ can be tiny if m scales subexponentially with N .
Quasi-go theorem.Next we study quantum computing with quasi codes, and we show that there exist quasi codes that overcome the limitation by Eastin-Knill no-go theorem (see Appendix 3), which shall be replaced by a quasi-go theorem.
We first study the case of exact encoding and quasi error correction, and the case of quasi encoding is deferred to Appendix 4. Transversal gates are of the form for j as the index of subsystems.The set of transversal logical gates is a Lie group, denoted as G [7].The effect of quasi error correction is an equivalence relation of logical gates, i.e., the gate-cell structure.
Theorem 2. There exist codes with exact encoding and quasi error correction such that the connected component of identity G 0 ⊂ G does not collapse to logical identity.
Proof.Let e iξD ∈ G 0 , ξ ∈ R then D = j α j H j for H j acting on the subsystem j, and DP = P DP [7].Given the error set {E jl } on block j, H j can be written as As P E jl P = e jl P + P B jl P from quasi error correction, then for h := jl α j β jl e jl , B := jl α j β jl B jl .We see that the size of P BP is in the order O(poly(N, d q , β)).
The terms D n P can be computed, and we find D n P ≈ n m=0 h m P (BP ) n−m .Thus, e iξD P ≈ e iξh e iξP BP P.
The unitary operator e iξP BP may be nontrivial for large ξ.If ξ ∈ o(1), then e iξD falls in the cell of logical identity.If ξ ∈ O(1) or ξ ≫ 1, then e iξD has a nontrivial logical action.
Our theorems can also be extended to subsystem quasi codes, see Appendix 5. We see that the gate-cell structure due to quasi codes prevent G 0 collapses to identity.Instead, G 0 will split into a collection of cells of various sizes.All gates in the same cell are equivalent with respect to the quasi codes.
As the group G is compact for a finite system size, the number of logical operators, i.e., quasi or exact gates, would be finite.This is not a problem for quasi universality since an infinite number of logical operators is not required.Furthermore, when the quasi code tends to be exact, which can be achieved in a certain limit, e.g., the large-N or large-d q limit, the sizes of gate-cells become smaller, and the number of logical operators could tend to be infinite.This is in sharp distinction from any exact quantum code for which the number of logical gates is fixed.Our proof demonstrates that quasi universality can be achieved in principle given a certain quasi code.We will see below that the SU (d)-covariant quasi codes achieve this, while the sizes of gate-cell for all gates are of the same order.SU (d)-covariant quasi codes.We find classes of covariant codes can be defined from matrix-product states [51,52] with global continuous symmetry, namely, the highersymmetry generalizations of VBS states [30][31][32][33], which have SPT order and appear as ground states of 1D local Hamiltonians.Below we define covariant codes with unitary symmetry, while the construction can be easily generalized to other Lie groups such as orthogonal and symplectic groups, and also to higher spatial dimensions.
The SU (d)-covariant codes we consider are defined by the isometry V : |α → |ψ α for |α ∈ H L and with the logical space H L as the fundamental representation of SU (d), and the physical space and each H n as the adjoint representation of the group SU (d).The N sites H n are known as 'bulk', and the site H L as 'edge'.Both the edge and bulk states are employed for the codes.The tensors A i n (n = 1, . . ., N ) are translation-invariant (hence n can be omitted) taking the form for [34,35].The channel E can be dilated to a unitary operator W such that W |0 = i A i |i , which is an isometry.The encoding isometry V is from the product of W , each as the dilation of E. From the global symmetry, it is clear that for N factors of U (g) acting on the bulk and Ȗ (g) acting on the edge, for g ∈ SU (d).Now we study QEC property of the codes.We label the edge as the (N +1)th site, and others from 1 to N sequentially.The edge before encoding can be viewed as the 0th site.The local state of the edge is This means that partial information of the logical state can be read off from the edge state via t k αα , yet this is exponentially suppressed when N increases, and σ N +1 → 1/d, which is the unique fixed point of E. The channel E has other eigenvalues as which is the complementary state of It is easy to see ρ n converges to the completely mixed state exponentially, ρ n → 1/(d 2 − 1) as n increases, yet for small n, the local state ρ n contains the observable t k αα of the logical state |α .This manifests that the codes are indeed quasi codes.
To define a proper error model, observe that the action of a local operator T a , as the adjoint representation of t a , can be converted to actions on the edge.The action of T a on a local site is T a i A i |i = i [A i , t a ]|i .Denote the link (n, n ± 1) as n±, then the action of T a n is a superposition of the actions of t a n+ and t a n− on the edge space.So we could view the information encoded in the bulk equivalently as encoded in the links, i.e., the history states of the edge.
For the local error set {t a n+ }, we find and also ψ α |t a n− |ψ β = χ n−1 t a αβ .This means local errors are only approximately detected.When d → ∞, or n → ∞, the detection becomes exact.For correlation functions we find for n > m ≥ 0, h bac = d bac + if bac , and d bac are also structure constants of SU (d).We see that this correlation function contains information t c αβ of the logical states, which is suppressed for large n or large d.In addition, this leads to We see that the edge is more correlated to bulk sites that are close to it, and there are exponential decay of bulk correlation functions.
The quasi conditions (19) and ( 20) also apply to errors in the algebra of {t a n+ } with t 0 ≡ 1.It is easy to see that a logical error gate resulting from an error on the link n+ depends on n, and in general can be expressed as a unitary operator E n = e iχ n k ǫ k t k , ǫ k ∈ R. We shall average over random errors that occur fairly on the system, and the net effect is a random unitary channel which, to the order O(χ 2n ) for each n, can be approximated by a unitary operator e iη k ǫ k t k for which vanishes when N → ∞ or d → ∞.The error analysis above is consistent with the erasure error model, for which the performance of covariant codes has been studied [46].From (19), we find If the environment can erase a local site n ∈ [1, N + 1] randomly, then it can read out the logical value t a αβ from a global observable.Then the uncorrectable part of the code is lower bounded by a quantity proportional to (N max n ∆T n ) −1 (also see Appendix 2), for ∆T n as the spectral range of a local observable T n , and max n ∆T n scales with the dimension d 2 − 1, which agrees with the scaling of η (22).
When the codes are prepared as ground states of frustration-free local Hamiltonians [30][31][32][33], which in general take the form H = n h n,n+1 , the nearest-neighbor interaction terms h n,n+1 , although do not commute with each other, play similar roles with stabilizers [53].Each term h n,n+1 shall be minimized for the code space C.An error E n will increase the energy term h n,n+1 , and can be corrected by cooling back to C.
For quantum computing with error correction, the sequence of gate operations is L ℓ=1 U ℓ E ℓ with the enacted logical gates U ℓ interrupted by error gates E ℓ .Product of U ℓ can yield the whole group SU (d).Recall that E ℓ depends on η, and when d → ∞, or N → ∞, then E ℓ → 1.For finite d and N , when the length L of the computation increases, the errors will increase, too.As we already know, the quasi feature limits the accuracy of logical gates.The parameter η can be viewed as the unique measure of the accuracy of logical gates, although each error gate E ℓ can be different in practice.Therefore, any logical gate is actually a gate-cell of size measured by η, and logical gates within this accuracy are equivalent.A SU (d)-covariant code defined above is quasi universal for SU (d), and it becomes exact universal when the code itself approaches exact.
Conclusion.In this work, we have introduced quantum computing with quasi-exact quantum codes, i.e., codes that can become exact under a certain limit.The covariant codes can provide a quasi-continuous universal and transversal set of logical gates.We note that by relaxing the requirement of covariance and allowing various sizes of gate-cells, this will naturally lead to the setting when a discrete set of universal gates is first realized, and all other gates are from product of them.It could be nontrivial to find a quasi code that can satisfy the (quasi-)universality, transversality, and discreteness of gate set simultaneously.
A slightly easier task is to employ concatenation of codes such that their transversal sets of gates can be combined to be universal.The concatenation can also benefit to increase the allowed error threshold.A quasi code can be taken as either an inner code or an outer code, and it can be concatenated with another quasi code or exact code, such as stabilizer codes.It is thus also important to see if there are quasi codes that permit universality with such concatenated transversality.
Acknowledgement.This work has been funded by grants from NSERC.

APPENDIX 1. Approximate quantum error correction
According to [26], a channel N on a code P is ǫcorrectable iff .The channels D and B maps system states to 'environment' states since {|i } are states of an environment, E. Furthermore, there exists a recovery channel R such that d(RN , 1) ≤ ǫ.The existence of a near-optimal recovery channel is also proved, but it may require a numerical convex optimization to find it [26].
The ǫ-correctability of the set {E i } does not guarantee that for other operators in its algebra, however.Namely, suppose another set of error operators {F ℓ } is defined by a matrix Υ = ℓi y ℓi |ℓ i| such that F ℓ = i y ℓi E i .Then the superoperator Υ defined by conjugation of Υ induces two new channels D = Υ • D and B = Υ • B to the condition (A1) for the new set {F ℓ }.As Υ is not contractive in general, namely, it does not necessarily reduce distance between states, the distance d( D + B, D) is not upper bounded by ǫ anymore.This poses a problem when considering the approximate correction of arbitrary local errors, and instead motivates the introduction of quasi codes.

Quasi codes
Here we include several codes defined in literature as examples of quasi codes.
Example 1 (U (1)-covariant codes [45][46][47]).Consider the truncated version of the three-rotor code [46] with the encoding Nǫ TL for a small angle 2π Nǫ that depends on ǫ, which is also the accuracy of the logical gates.A gatecell structure shall be introduced, then the U (1) group reduces to a finite group Z ⌊Nǫ⌋ .When ǫ → 0, i.e., m → ∞, the size of the gate-cell shrinks, and the number of gate-cells increases, so Z ⌊Nǫ⌋ → U (1).This agrees with our notation of quasi-universality.
Example 2 (Cat codes [56][57][58]).The class of cat codes employs coherent states of photons, or bosons in general, and their superposition for the encoding.For instance, a logical qubit can be encoded as The errors that are natural for cat codes include loss or gain by a certain orders of â or â † , and dephasing errors of the form e iθ n.The violation of the condition ( 1) is typically by a term of order O(α c1 e −c2α c 3 ), with constants c 1 , c 2 , c 3 [58].If the code becomes exact, i.e., for |α| → ∞, the error correction also becomes exact.
Example 3 (Bosonic SPT codes [53,59,60]).Qubits are defined for 1D spin chains with bosonic SPT order [53].The logical space is the degenerate ground space due to symmetry breaking.The logical bit-flip X L is the generator of the broken symmetry, and logical phase-flip Z L is the SPT order parameter.
We find in general the codes are quasi exact.For instance, for Majumdar-Ghosh model of spin-half chain [61,62], the ground states are product of singlets |s n,n+1 on nearest-neighbor (nn) sites, and the codewords are for N + 1 ≡ 1.They have overlap e −N/ξ which vanishes when the system size N → ∞, for a constant correlation length ξ.Furthermore, the error-correction is also quasiexact.The spin correlation function is a step function: it is 1 for nn sites, and 0 for non-nn sites.This means that it exactly corrects local errors for non-nn sites, but cannot correct local errors on nn sites.For higher-spin chains, spin correlation function falls off exponentially [59].Errors that are far away with spatial distance r are quasi correctable with exponentially small non-correctable part O(e −r/ξ ).To enforce this, it is not enough to make the system size big.Instead, the system has to be cooled down to temperatures lower than the gap to make sure that mostly errors are dilute and would not occur on nnsites.
Example 4 (Anyons [17]).Qubits can be encoded in the fusion space of multiple non-Abelian anyons, the braiding of which can enable universal quantum computing.Quantum gates are from the braid group restricted on the code space.We find that qubits encoded by non-Abelian anyons in non-exactly solvable models is an example of quasi code.For a system of anyons, the logical gates are from braidings which are homologically nontrivial.Given a set of local errors {E i } such that the support of E † i E j for any i, j is homologically trivial, then for a set of logical states {|ψ n }, it satisfies and the overlap ψ n |ψ m → 0 when the system size N → ∞, ψ n |B ij |ψ m ∈ O(e −r/ξ ) for r as the typical separation of anyons.When all separations among anyons become large, the codes and error-correction become exact.The quantum gates from braiding can be viewed as non-Abelian geometric phases.When nonabelian anyons are close together, their different fusion channels are split in energy, then braiding will induce dynamical phase errors besides geometric phases.
For instance, a qubit can be encoded by 1D Majorana wire [63] with Majorana zero modes (MZM) at each end.Two MZM form a fermion, for which a basis is formed by the occupied |1 and unoccupied |0 states.The states have overlap e −N/ξ which vanishes when the separation between the modes N → ∞, given a finite correlation length ξ.Therefore, MZM qubit is a quasi code.
The braiding operation is typically not transversal, though.On a spin-lattice model, a braiding operation typically requires a linear-depth local unitary circuit [18][19][20][21][22]64].As such, it may spread out a local error which can further induce a logical error.Careful error-corrections are needed during and after the braiding operations.MZM, as defects in 1D fermionic SPT order, is an Ising anyon.It has been well studied that braiding of MZM may spread out local errors [65].

Eastin-Knill no-go theorem
We review the contents of Eastin-Knill no-go theorem [7], which basically states that there is no universal set of transversal logical gates (TLG) supported by a finite-dimensional physical system.
A unitary operator U is a logical operator iff U P = P U P. (A6) A unitary operator U † is a logical operator iff P U = P U P .It turns out, if U is logical, then U † is also logical.This means a state in the code space cannot be mapped out of it by U , while a state out of the code space cannot be mapped into it by U , neither.It is also equivalent to say [U, P ] = 0.If the code space is defined by a Hamiltonian H, then it has [P, H] = 0.If the codewords are further degenerate, then HP = hP for a constant h which can be set to zero.A logical operator U can be viewed as an 'emergent' symmetry of H as P [U, H]P = 0, which means U preserves H on the code space P .While in general U is not a symmetry of H.As a result, we see that emergent symmetry plays more important roles than symmetry in the presence of a Hamiltonian for logical operations.
In order to define transversal gates, we need to introduce 'transversal part', or subsystem.An errorcorrection subsystem, or subsystem for short, is a part of the whole system for which error correction can be performed.The natural choice of a subsystem is a local site.However, a subsystem can consist of several local sites, which should usually be a connected local part of the whole system.This applies to topological codes that have macroscopic code distances.
A TLG does not couple different non-overlapping subsystems within the same logical qubit, and only couples a corresponding subsystem from different logical qubits.TLG do not spread out errors across subsystems for the same logical qubit.So the error-correction for each logical qubit can ensure fault tolerance.
For many logical qubits each encoded by a different physical system, denoted as Q[r], usually a one-to-one correspondence of subsystems has to be chosen.For instance, for two logical qubits Q [1] can be chosen to correspond to Q n [2] for the same label n.In general, there might be a permutation π(n).If each system comes with a Hamiltonian H[r], then the total Hamiltonian is the sum of them H = r H[r] without interaction terms.
A transversal gate takes the form for j as the index of subsystems.Note that U is defined up to any permutation of subsystems since permutation cannot spread out errors except the locations of them.
In order to see the generality of Eastin-Knill theorem, below we review its content in details.There are several crucial assumptions.(a) The error set {E i } spans the space of a subsystem, namely, it requires arbitrary errors on a subsystem can be detected.If the dimension of the space spanned by {E i } is smaller that that of a subsystem, the theorem does not apply.(b) Transversality is fixed, namely, all logical transversal gate takes the form (A7) for a given partition of subsystems.(c) The Hilbert space dimension of the system is finite.(d) The universality is exact, i.e., the group SU (d) shall be realized efficiently.
Given a code space P , it first shows that the set of logical gates (A6) form a Lie group L. Given a transversality and the form (A7), it shows that the set of TLG is also a Lie group G = L A, for A = j U (d j ), d j as the dimension of a subsystem.Now the connected component of identity G 0 in G contains elements of the form An operator e iξD is a TLG for ξ ∈ R, and then it shows DP = P DP .The operator D can be written as a sum of local terms D = j α j H j due to the structure of Lie algebra, and each H j acts on the subsystem j.Given the detection of arbitrary local errors, it holds P H j P ∝ P , and then DP = P DP ∝ P .As the result, CP ∝ P , which means that C acts as the logical identity gate, and the whole group G 0 'collapses' to identity.As the quotient group Q = G/G 0 is a topologically discrete group, the number of logically distinct operators is finite.In other words, the set of transversal logical gates is not universal.
Next we remark on some points.(1) During the execution of each logical gate, there may be leakage out of the code space, as long as it goes back to the code space at the end.(2) Each local unitary U j can be realized in many ways, even not unitarily, as long as the net effect is unitary.For instance, ancilla and measurement can be used.(3) There is no logical ancilla to realize a logical gate U since U itself must be unitary of the form (A7). (4) A subsystem can contain several local sites, and this especially applies to codes with large code distance, such as topological codes.Error correction on local sites ensures error correction on a subsystem.For code distance d = 2t + 1, a subsystem can be as big as t.This means that each U j can be an entangling gate on the underlying local sites.However, the transversality has to be fixed to ensure that all logical gates are of the form (A7). ( 5) The theorem applies to arbitrarily large but finite dimensional Hilbert space.This fact is crucial to generalize it to the quasi-exact setting, e.g., the group-theoretic argument still applies.

Quasi-go theorem
For the case of quasi encoding, P 2 = P , there will be further perturbations from the encoding itself in addition to perturbations by the quasi QEC.We have to consider quasi gates which becomes exact when the encoding becomes exact, i.e., δ → 0. A quasi gate U is defined such that U P := P U P + L U P, P U := P U P + P R U , (A8) and both L U P and P R U are small and tend to zero when the encoding becomes exact.From P 2 = P + K, K = V ∆V † , we see that L 1 P = P R 1 = −K.For product of two gates U and V , the difference between U V P and P U V P will increase.Different from the exact case, the set of quasi gates, denoted by G δ , is an open subset of G instead of a Lie group.When δ → 0, G δ → G.We cannot employ the argument for exact encodings directly, but we can still employ it in the following way.A gate of the form C = e iξD can be a quasi gate, quasi identity gate, or not be a gate, and only the first case is nontrivial.Suppose there exists a quasi gate of the form e iξD for a certain value of ξ and D = j α j H j .For other values of ξ, e iξD may not be a quasi gate.Let e iξD P = P e iξD P + L ξ P , ∀ξ ∈ R, and L ξ may not be a smooth function of ξ and may not be bounded.We find If L ξ is differentiable for all ξ, then A = −i∂ ξ L ξ P , which is not a zero operator in general.With quasi error correction, we find the difference for a certain parameter ε, operators A and B. As a result, the operator e iξD P is a nontrivial quasi gate depending on A and B. This means that gates of the form e iξD are logically different.

Subsystem quasi codes
Here we extend the results above to the subsystem quasi codes, which are the quasi version of subsystem codes.In this case, the code space C has a tensor product form for the truly logical subspace T that encodes the information and a junk 'gauge subspace' J that does not encode logical information.The apparent encoding operator is now not one-to-one V : |i → |ψ i |ψ j , for |i ∈ H L , |ψ i ∈ T , and any state |ψ j ∈ J (which can also be mixed states), but it is effectively one-to-one when the J part is ignored.The total space takes the form for S as the syndrome subspace.For error correction, in additional to leakage to S, errors may also generate entanglement between T and J .A good subsystem code is designed such that J can benefit the error correction.We still use P as the exact or quasi projector on C, use P T as the exact or quasi projector on T , while use P J as the exact projector on J .We assume an orthonormal basis of J can be chosen to define P J .
The exact error correction for a set of errors {E i } on subsystem codes has been shown [66][67][68] to be P E † i E j P = (1 T ⊗ J ij )P, (A13) for 1 T (J ij ) acting on the space T (J ).Contrary to the standard condition (1), here there can be a nontrivial action J ij on the gauge part.The condition above is equivalent to for E ig := E i |ψ g , with any |ψ g ∈ J .The effective error operators E ig are rectangular, namely, they can map states in T to states in T and J .The span of {E ig } is the product of the span of {E i } and {|ψ g }.If the dimension of J is d J , then a set of d 2 J linearly independent states {|ψ g } is enough to ensure the correction of other errors in the span of {E ig }.The recovery channel R will map states in T and J (also S) back to states in T .In other words, we see that the modification to the standard condition (1) is to replace square error operators by rectangular ones, and increase the number of them by a factor of d 2 J .We shall denote the set {E ig } as {E I } for the new label I := ig.
Our Theorem 1 can be generalized to subsystem quasi codes with the replacement of {E i } by {E I }, P by P T , and σ C by σ T .Now dimE is also polynomial of d J .
To ensure the infidelity is small, the size of the noncorrectable part may or may not depend on d J as long as the dependence of N or d q guarantees the smallness of infidelity.
Unitary logical operators U on subsystem codes not only commute with P , but also factorize as U P = (U T ⊗ 1 J )P, (A15) for a nontrivial gate U T .The Eastin-Knill theorem holds for subsystem codes, while note that in the proof the projector is P instead of P T , and the actions on the gauge part are ignored.Theorem 2 can be extended to subsystem quasi codes, while replacing h by a nontrivial matrix on J .There still could be a nontrivial logical gate e iξP BP on the code subspace T .Finally, with quasi encoding the operator K needs to be replaced by K ⊗ P J , and with the additional condition (A15) on quasi gates, operators of the form e iξD P can be nontrivial quasi gates on subsystem quasi codes due to (A10).