Charge-conserving unitaries typically generate optimal covariant quantum error-correcting codes
MMIT-CTP/5290
Charge-conserving unitaries typically generate optimal covariantquantum error-correcting codes
Linghang Kong ∗ and Zi-Wen Liu † Center for Theoretical Physics, MIT, Cambridge, MA 02139, United States Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
Abstract
Quantum error correction and symmetries play central roles in quantum information scienceand physics. It is known that quantum error-correcting codes covariant with respect to continu-ous symmetries cannot correct erasure errors perfectly (an important case being the Eastin–Knilltheorem), in contrast to the case without symmetry constraints. Furthermore, there are funda-mental limits on the accuracy of such covariant codes for approximate quantum error correction.Here, we consider the quantum error correction capability of random covariant codes. In partic-ular, we show that U (1)-covariant codes generated by Haar random U (1)-symmetric unitaries,i.e. unitaries that commute with the charge operator (or conserve the charge), typically saturatethe fundamental limits to leading order in terms of both the average- and worst-case purifieddistances against erasure noise. We note that the results hold for symmetric variants of uni-tary 2-designs, and comment on the convergence problem of charge-conserving random circuits.Our results not only indicate (potentially efficient) randomized constructions of optimal U (1)-covariant codes, but also reveal fundamental properties of random charge-conserving unitaries,which may underlie important models of complex quantum systems in wide-ranging physicalscenarios where conservation laws are present, such as black holes and many-body spin systems. One of the most fundamental and widely studied ideas in quantum information processing is quan-tum error correction [1, 2, 3, 4], which protects (logical) quantum systems against noise and errorsby suitably encoding them into quantum codes living in a larger physical Hilbert space. Besidesthe clear importance to the practical realization of quantum computing and other technologies,quantum error correction and quantum codes have also drawn great interest in physics recently asthey are found to arise in many important physical scenarios in e.g. holographic quantum gravity[5, 6] and many-body physics [7, 8, 9].Physical systems typically entail certain symmetries or conservation laws, which have played centralroles in the developments of many areas in physics. Given the broad practical and physical relevance ∗ [email protected] † [email protected] a r X i v : . [ qu a n t - ph ] F e b f quantum error correction, it is important to understand its performance and limitations undersymmetry constraints. More explicitly, the encoders are restricted to be covariant with respect tosome symmetry group (i.e. commute with certain group actions), generating the so-called covariantcodes [10, 11, 12]. Covariant codes are known to have broad relevance in both practical andtheoretical aspects, arising in many important areas in quantum information and physics such asfault tolerance [13], quantum reference frames [10], AdS/CFT correspondence [14, 15, 16, 11, 12],and condensed matter physics [9].When the symmetry group is continuous, there exists fundamental limitations on the error cor-rection capability of the corresponding covariant codes. The Eastin–Knill theorem [13] is a wellknown no-go theorem in this regard, which indicates that codes covariant with respect to contin-uous symmetries in the sense that the logical group actions are mapped to “transversal” physicalactions that are tensor product on physical subsystems (a feature favorable for fault tolerance forthat they do not spread errors on one subsystem to others) cannot correct local errors perfectly (forphysical system with finite Hilbert space dimension). Another understanding of this phenomenonis that some logical information is necessarily leaked into the environment due to the error, whichforbids perfect recovery. However, it is still possible to perform error correction approximately.Several “robust” versions of the Eastin-Knill theorem or lower bounds on the inaccuracy of covari-ant codes are recently found [11, 12, 17, 18, 19], some of which employing methods from other areasof independent interest such as quantum clocks [12], quantum metrology [17, 18, 19], and quantumresource theory [18, 20].This work concerns the achievability of such lower bounds. In particular, here we consider thesimple but important case of U (1) symmetry corresponding to charge (energy) conservation, whichis ubiquitous in physical systems. We analyze codes generated by unitaries drawn from the Haarmeasure that obey the charge conservation law, i.e. commute with the charge operator, which areparticularly interesting because: i) The results indicate typical properties of all charge-conservingunitaries due to the Haar measure; ii) Haar random unitaries and their relatives including designsand random circuits play essential roles in the study of many-body quantum systems such as blackholes [21, 22, 23] and chaotic spin systems [24, 25], indicating that our refined model with conservedquantities is potentially of broad interest in physics. We rigorously analyze the performance of ourrandomly constructed covariant code against erasure noise, as characterized by both the average-case and worst-case error (measure by the purified distance) among different input states. To do so,we use the complementary channel technique [26], which gives characterization of the error rate of acode by the amount of information leaked into the environment. For our random code we essentiallybreak down the error into two components, one characterizing the deviation of the random codefrom its average leading to error which can be bounded using a “partial decoupling” theorem[27] and turns out to be exponentially small, while the other characterizing a polynomially smallintrinsic error given by the average state. We show that in certain important cases our randomcode almost always saturates the lower bounds in [11] to leading order, indicating that charge-conserving unitaries typically give rise to optimal covariant codes. The results hold if the Haarrandom charge-conserving unitary is simplified to corresponding 2-designs, and as we conjecture,efficient random circuits composed of charge-conserving local gates. Note that in our case withcharge conservation the error is intrinsically polynomially small, while in the no-symmetry case theerror of such Haar random codes is normally exponentially small and there are perfect codes.2he rest of this paper is structured as follows. In Section 2 we formally introduce the relevantbackground. In Section 3 we describe the random code construction, and show that the codes areindeed covariant. In Section 4, we present rigorous analysis of the error of our random code asmeasured by Choi and worst case purified distances and make a few comments about the noisemodel and input charge. In Section 5, we explicitly compare our upper bounds on the error ofour random code with known lower bounds. In Section 6, we discuss the extension of our study tosymmetric t -designs and random circuits as well as several associated key problems. We concludethe work with relevant discussions in Section 7. Several detailed technical calculations are left tothe appendices. Here, we formally introduce the definitions and techniques that play key roles in this work.
The closeness of two quantum states ρ and σ can be characterized by their fidelity F ( ρ, σ ) ≡ (cid:107)√ ρ √ σ (cid:107) = Tr (cid:113) √ ρσ √ ρ. (1)Note that the fidelity is sometimes defined as F ( ρ, σ ) in the literature, but we shall stick to thedefinition of Eq. (1) in this work. The purified distance P ( ρ, σ ) is then defined as P ( ρ, σ ) = (cid:112) − F ( ρ, σ ) . It is known [28, Section II] that the purified distance satisfies the triangle inequality P ( ρ, σ ) ≤ P ( ρ, τ ) + P ( τ, σ )for any state τ . It satisfies the following relation with 1-norm distance [2]12 (cid:107) ρ − σ (cid:107) ≤ P ( ρ, σ ) ≤ (cid:112) (cid:107) ρ − σ (cid:107) . (2)In this work, we consider approximate quantum error correction, whose performance is quantifiedby comparing the input state and the state obtained after error correction. The input state is ajoint state on the logical system L and a reference system R . Let E be the encoding channel ofsome error correction code, and N be the noisy channel.To characterize the overall performance of the code, we consider two widely used strategies. Thefirst one makes use of the Choi isomorphism, which essentially considers a maximally entangledstate as the input and characterizes the average-case error. More explicitly, when the referencesystem R has the same Hilbert space dimension as L , we define the Choi fidelity and Choi error ofthe code as F Choi ≡ max D F ( ˆ φ LR , [( D ◦ N ◦ E ) L ⊗ id R ]( ˆ φ LR )) , (cid:15) Choi = (cid:113) − F , φ LR is the maximally entangled state between L and R ,ˆ φ LR = | ˆ φ (cid:105)(cid:104) ˆ φ | LR , | ˆ φ (cid:105) LR = 1 √ d L d L − (cid:88) i =0 | i (cid:105) L | i (cid:105) R , and d L is the Hilbert space dimension of L . Note that this characterization of error via Choistates is closely related to the average behavior in the following sense. It is natural to define theaverage-case error as (cid:15) A := (cid:90) dψP ( ψ, ( D ◦ N ◦ E ) ψ ) , where ψ L denotes a pure logical state and the integral is over the uniform Haar measure. Then wehave that (cid:15) Choi and (cid:15) A are related by (cid:15) Choi = (cid:114) d + 1 d (cid:15) A , where d is the dimension of the input (logical) system [29, 30]. In particular, as d increases (cid:15) Choi and (cid:15) A approach the same value. The second one is based on considering the worst-case fidelityand worst-case error defined as F worst ≡ max D min R,ρ LR F ( ρ LR , [( D ◦ N ◦ E ) L ⊗ id R ]( ρ LR )) , (cid:15) worst = (cid:113) − F . Note that the minimization runs over all reference systems R and all input states ρ LR .The code errors (cid:15) Choi and (cid:15) worst could be characterized using the formalism of complementarychannels [26]. Let A be the intermediate system that N ◦ E maps to. It is always possible to view(
N ◦ E ) L → A as a unitary mapping from L to the joint system A and the environment E , followedby a partial trace over E . The complementary channel ( (cid:92) N ◦ E ) L → E is given by tracing over A andoutputs the state left in the environment. Intuitively, an encoding is good if the environment doesnot learn much about the input. To be more precise, (cid:15) Choi = min ζ P (( (cid:92) N ◦ E L → E ⊗ id R )( | ˆ φ (cid:105)(cid:104) ˆ φ | LR ) , ( T L → Eζ ⊗ id R )( | ˆ φ (cid:105)(cid:104) ˆ φ | LR )) , (3) (cid:15) worst = min ζ max ρ LR P (( (cid:92) N ◦ E L → E ⊗ id R )( ρ LR ) , ( T L → Eζ ⊗ id R )( ρ LR )) , (4)where T ζ is the constant channel T L → Eζ ( ρ L ) = Tr[ ρ L ] ζ E . A property of the constant channel is that( T L → Eζ ⊗ id R )( ρ LR ) = ζ E ⊗ Tr L [ ρ LR ] , (5)which will be useful for our calculations later. 4 .2 Covariant codes Let G be a Lie group, and let g → U A ( g ) and g → U L ( g ) be representations of G on the physical andlogical Hilbert spaces respectively. A code is covariant with respect to G if the encoding channel E commutes with the representations, i.e. E ( U L ( g ) ρU L ( g ) † ) = U A ( g ) E ( ρ ) U A ( g ) † for all g ∈ G and state ρ . A standard scenario (consider the Eastin–Knill theorem for local errors)is when U A ( g ) takes the tensor product (transversal) form U A ( g ) = U ( g ) ⊗ U ( g ) ⊗ . . . ⊗ U n ( g ) , where U i ( g ) acts on the i -th physical subsystem. The generators on the group G will be representedas T L and T A on the logical and physical Hilbert spaces, and the tensor product form dictates that T A takes the form T A = n (cid:88) i =1 ( T A ) i , where ( T A ) i only acts on i -th qubit. For a bipartite state ρ P Q , the conditional min-entropy (conditioned on Q ) is defined as H min ( P | Q ) ρ = sup σ ≥ , Tr σ =1 sup { λ ∈ R | − λ I P ⊗ σ Q ≥ ρ P Q } . For pure state ψ = | ψ (cid:105)(cid:104) ψ | P Q , there is a simple formula for H min ( P | Q ) ψ . Let the Schmidt coefficientsof | ψ (cid:105) be α , . . . , α D , then the conditional min-entropy is given by H min ( P | Q ) ψ = − α + . . . + α D ) . (6)To see this, note that [31] for any tripartite pure state ρ on X, Y and Z , H min ( X | Y ) ρ + H max ( X | Z ) ρ = 0 . where the conditional max-entropy H max is defined as H max ( X | Z ) ρ = sup σ Z log F ( ρ XZ , I X ⊗ σ Z ) . Note that I X ⊗ σ Z is not a normalized quantum state and F ( ρ XZ , I X ⊗ σ Z ) should be interpretedas d X F (cid:16) ρ XZ , I X d X ⊗ σ Z (cid:17) .In our case the state | ψ (cid:105) is pure on P and Q , so we can choose the third register R to be a trivialsystem, and therefore H min ( P | Q ) ψ = − H max ( P | R ) ψ = − (cid:107) (cid:112) ψ P √ I P (cid:107) = − (cid:112) ψ P ]) = − α + . . . + α D ) . (7)5 .4 Decoupling and partial decoupling The (one-shot) decoupling theorem [32] charaterizes the degree to which a system is decoupled fromthe environment under certain channels in terms of (suitable variants of) conditional min-entropies.It can actually be viewed as a concentration-of-measure type bound where the randomness comesfrom a Haar random unitary acting on the system. To be more precise, for any bipartite state ρ AR and quantum channel τ A → E , the decoupling theorem gives an upper bound for the followingquantity E U A ∼ Haar (cid:107) ( T A → E ⊗ id B )[ U A ρ AR U A † ] − ( T A → E ⊗ id B )[ ρ AR avg ] (cid:107) where ρ AR avg ≡ E U A ∼ Haar U A ρ AR U A † . A generalization of decoupling called partial decoupling that will be useful for our purpose wasstudied in [27], where the unitary U A could take a more general form. We assume that the Hilbertspace of A takes the form of a direct-sum-product decomposition H A = J (cid:77) j =1 H A l j ⊗ H A r j , and U A satisfies U A = J (cid:77) j =1 I A l j ⊗ U A r j where U A r j is Haar random within H A r j . The distribution of such U A will be called H × . Let l j and r j be the dimensions of H A l j and H A r j respectively. The (non-smoothed) partial decoupling theorem[27, Eq. (79)] states that E U ∼ H x (cid:104)(cid:13)(cid:13) T A → E ◦ U A (cid:0) Ψ AR (cid:1) − T A → E (cid:0) Ψ AR avg (cid:1)(cid:13)(cid:13) (cid:105) ≤ − H min ( A ∗ | RE ) Λ(Ψ , T ) . Here, the state Λ(Ψ , T ) is defined asΛ(Ψ , T ) = F (Ψ AR ⊗ τ ¯ AE ) F † , where τ ¯ AE is the Choi-Jamiolkowski state of T and the operator F A ¯ A → A ∗ is F A ¯ A → A ∗ := J (cid:77) j =1 (cid:115) d A l j r j (cid:104) Φ lj | A l ¯ A l (cid:16) Π Aj ⊗ Π ¯ Aj (cid:17) with | Φ lj (cid:105) being the maximally mixed state. Π Aj is the projector into H A l j ⊗ H A r j . Again the averagestate is defined as Ψ AR avg = E U ∼ H × U Ψ AR U † . and could be calculated using the formulaΨ AR avg = J (cid:77) j =1 Ψ A l Rjj ⊗ I A r j r j , Ψ A l Rjj = Tr A r [Π Aj Ψ AR Π Aj ] . (8)6 Covariant codes from random unitaries
We now define the construction of U (1)-covariant quantum error-correcting codes based on randomcharge-conserving unitaries that will be analyzed in this work.Consider the Hamming weight operator on m qubits Q ( m ) = m (cid:88) i =1 I − Z i , where Z = | (cid:105)(cid:104) | − | (cid:105)(cid:104) | is the Pauli Z operator on a single qubit, and Z i is the operator Z actingon qubit i . We consider codes that encode k logical qubits in n physical qubits, that are requiredto be covariant with respect to U (1) represented by (without loss of generality) Hamming weightoperator. To be more precise, we consider Lie group U (1), where the group action e iθ ∈ U (1) isrepresented as e iθ → e iθQ ( k ) and e iθ → e iθQ ( n ) on the logical and physical Hilbert spaces respectively.Our construction relies on n -qubit unitaries that conserve the Hamming weight, i.e. commute with Q ( n ) . Such unitaries are block diagonal with respect to the eigenspaces of Q ( n ) . Let H × be theHaar measure on the group of such unitaries. We define ( n, k, α ) -codes as follows. Definition 1.
We call a code a ( n, k, α ) -code if it encodes k logical qubits in n physical qubits byfirst appending an ( n − k ) -qubit state | ψ α (cid:105) with Hamming weight α , i.e. Q ( n − k ) | ψ α (cid:105) = α | ψ α (cid:105) , and then applying a unitary U on the joint n -qubit system that commutes with Q ( n ) . In particular,a ( n, k, α )-random code is given by U drawn from H × . It is easy to verify that such codes indeed satisfy the covariance condition.
Proposition 1. ( n, k, α ) -codes are covariant with respect to a U (1) symmetry, with the logicalcharge operator T L = Q ( k ) and physical charge operator T A = Q ( n ) . Note that this property holdsfor the ( n, k, α ) -random code.Proof. Since the n -qubit unitary U commutes with Q ( n ) , it commutes with e iQ ( n ) θ for all θ . Thenfor any k -qubit logical state ρ we have e iQ ( n ) θ U ( ρ ⊗ | ψ α (cid:105)(cid:104) ψ α | ) U † e − iQ ( n ) θ = U e iQ ( n ) θ ( ρ ⊗ | ψ α (cid:105)(cid:104) ψ α | ) e − iQ ( n ) θ U † = U ( e iQ ( k ) θ ρe − iQ ( k ) θ ⊗ e iQ ( n − k ) θ | ψ α (cid:105)(cid:104) ψ α | e − iQ ( n − k ) θ ) U † = U ( e iQ ( k ) θ ρe − iQ ( k ) θ ⊗ | ψ α (cid:105)(cid:104) ψ α | ) U † , which means the encoding map E satisfies the covariance condition e iQ ( n ) θ E ( ρ ) e − iQ ( n ) θ = E ( e iQ ( k ) θ ρe − iQ ( k ) θ ) . Again, note that the ( n, k, α )-random code can be viewed as a randomized construction of covariantcode, and in addition, it reveals the average, or typical, properties of charge-conserving unitariesdue to the Haar measure. 7
Performance of random covariant codes
We now derive explicit bounds on both Choi and worst-case errors of ( n, k, α )-random code againsterasure of t qubits in terms of purified distance. We will fix the set of erased qubits in our analysis,but the results still hold when the t qubits are picked randomly, as will be discussed in Section 4.3. In this part we analyze the Choi error of ( n, k, α )-codes. Since the noise channel we consider isan erasure of t qubits, the complementary channel will be a partial trace over the n − t unaffectedqubits. We denote this by Tr n − t [ · ]. We obtain the following result. Theorem 1.
In the large n limit, when k and t satisfy k t = o ( n ) and α = an with < a < being a constant, the expected Choi error of the ( n, k, α ) -random code satisfies E (cid:15) Choi ≤ √ tk n (cid:112) a (1 − a ) (cid:18) O (cid:18) k t n (cid:19)(cid:19) . (9) Furthermore, the probability that the Choi error of a ( n, k, α ) -code (with respect to H × ) violates theinequality above is exponentially small, i.e. Pr (cid:34) (cid:15) Choi > √ tk n (cid:112) a (1 − a ) (cid:18) O (cid:18) k t n (cid:19)(cid:19)(cid:35) = e − Ω( n ) . (10) Proof.
By Eq. (4), for a specific choice of U in our ( n, k, α )-code construction, the Choi error ofthe corresponding code is given by (cid:15) Choi = min ζ P (cid:18) Tr n − t [ U ( | ˆ φ (cid:105)(cid:104) ˆ φ | ⊗ | ψ α (cid:105)(cid:104) ψ α | ) U † ] , I k ⊗ ζ (cid:19) ≤ P (cid:16) Tr n − t [ U ( | ˆ φ (cid:105)(cid:104) ˆ φ | ⊗ | ψ α (cid:105)(cid:104) ψ α | ) U † ] , Tr n − t Φ avg (cid:17) + min ζ P (cid:18) Tr n − t Φ avg , I k ⊗ ζ (cid:19) ≤ (cid:114) (cid:13)(cid:13)(cid:13) Tr n − t [ U ( | ˆ φ (cid:105)(cid:104) ˆ φ | ⊗ | ψ α (cid:105)(cid:104) ψ α | ) U † ] − Tr n − t Φ avg (cid:13)(cid:13)(cid:13) + min ζ P (cid:18) Tr n − t Φ avg , I k ⊗ ζ (cid:19) , (11)where the average state is Φ avg = E U ∼ H × U ( | ˆ φ (cid:105)(cid:104) ˆ φ | ⊗ | ψ α (cid:105)(cid:104) ψ α | ) U † . Note that Eq. (5) has been used in the first line, and the second line follows from the triangleinequality of P .When averaging over U sampled from H × , the first term in Eq. (11) can be bounded using the8artial decoupling theorem: E U ∼ H × (cid:114) (cid:13)(cid:13)(cid:13) Tr n − t [ U ( | ˆ φ (cid:105)(cid:104) ˆ φ | ⊗ | ψ α (cid:105)(cid:104) ψ α | ) U † ] − Tr n − t Φ avg (cid:13)(cid:13)(cid:13) ≤ (cid:114) E U ∼ H × (cid:13)(cid:13)(cid:13) Tr n − t [ U ( | ˆ φ (cid:105)(cid:104) ˆ φ | ⊗ | ψ α (cid:105)(cid:104) ψ α | ) U † ] − Tr n − t Φ avg (cid:13)(cid:13)(cid:13) ≤√ × − H min ( A ∗ | RE ) Λ . We prove in Appendix A that H min ( A ∗ | RE ) Λ = Ω( n ) , which implies that the expectation value of the first term is exponentially small. A simple applica-tion of the Markov’s inequality shows that this term is exponentially small with probability equalto one minus an exponentially small amount.The second term in Eq. (11) is independent of U . Since this is a minimization, an upper bound onthis term can be found by any choice of ζ . We set ζ to be the t -qubit marginal state of Tr n − t [Φ avg ],and as detailed in Appendix B we obtain P (cid:18) Tr n − t Φ avg , I k ⊗ ζ (cid:19) ≤ √ tk n (cid:112) a (1 − a ) (cid:18) O (cid:18) t k n (cid:19)(cid:19) . Here we consider the worst-case error of ( n, k, α )-codes under erasure of t qubits. The followinglemma gives an lower bound of worst case purified distance for a fixed code. Lemma 1 ([11, Thm. 3]) . For any encoding channel E and any noise channel N , let (cid:92) N ◦ E be acomplementary channel of
N ◦ E . Fixing a basis of logical states {| x (cid:105)} , we define ρ x,x (cid:48) = (cid:92) N ◦ E ( | x (cid:105)(cid:104) x (cid:48) | ) . (12) Assume that there exists a state ζ , as well as constants (cid:15), (cid:15) (cid:48) > such that P ( ρ x,x , ζ ) ≤ (cid:15), (13) (cid:107) ρ x,x (cid:48) (cid:107) ≤ (cid:15) (cid:48) , ∀ x (cid:54) = x (cid:48) . (14) Then, the code E is an approximate error-correcting code with an approximation parameter satis-fying (cid:15) worst ≤ (cid:15) + d L √ (cid:15) (cid:48) , (15) where d L is the dimension of the logical system.If one of several noise channels is applied at random but it is known which one occurred, thenEq. (15) holds for the overall noise channel if the assumptions above are satisfied for each individualnoise channel. n, k, α )-code construction, ρ x,x (cid:48) = Tr n − t [ U ( | x (cid:105)(cid:104) x (cid:48) | ⊗ | ψ α (cid:105)(cid:104) ψ α | ) U † ] . Note that the lemma above applies to a fixed encoding E . To generalize this theorem to ourrandomized construction, we define ρ x,x (cid:48) avg as ρ x,x (cid:48) in Eq. (12) averaged over the random unitary in E . Then using the following lemma, we can obtain bounds on the worst-case error. Lemma 2.
Consider the large n limit where parameters α , k , t , (cid:15) , δ and δ (cid:48) each might depend on n . If the average states ρ x,x (cid:48) avg satisfy P ( ρ x,x avg , ζ ) ≤ (cid:15) for some fixed state ζ independent of x , thenwith probability at least − p − p the code generated by our construction satisfy (cid:15) worst ≤ (cid:15) + δ + 2 k √ δ (cid:48) . Here, p and p are defined as log p = k − n (cid:26) H (cid:16) αn (cid:17) , H (cid:18) α + kn (cid:19)(cid:27) + t δ + O (log n ) , log p =2 k − n (cid:26) H (cid:16) αn (cid:17) , H (cid:18) α + kn (cid:19)(cid:27) + t + log 1 δ (cid:48) + O (log n ) . (16) Proof.
We use the partial decoupling theorem to find an upper bound for the average distancebetween ρ x,x (cid:48) avg and ρ x,x (cid:48) . Then by Markov inequality this bounds the probability that ρ x,x (cid:48) behavesmuch worse than ρ x,x (cid:48) avg . Finally we can show the code has good performance with high probabilityusing a union bound.Given that there exists ζ such that P ( ρ x,x avg , ζ ) ≤ (cid:15) for all x , we have E U P ( ρ x,x , ζ ) ≤ P ( ρ x,x avg , ζ ) + E U P ( ρ x,x , ρ x,x avg ) ≤ (cid:15) + E U (cid:113) (cid:107) ρ x,x − ρ x,x avg (cid:107) ≤ (cid:15) + (cid:113) E U (cid:107) ρ x,x − ρ x,x avg (cid:107) ≤ (cid:15) + √ × − H x min . where H x min is H min ( A ∗ | RE ) Λ(Ψ , T ) with the initial state set to be | x (cid:105)(cid:104) x | . By Appendix C, H x min = nH (cid:18) | x | + αn (cid:19) − t + log( n ) ≥ n min (cid:26) H (cid:16) αn (cid:17) , H (cid:18) α + kn (cid:19)(cid:27) − t + O (log n ) , which means that for each x ,log Pr[ P ( ρ x,x , ζ ) ≥ (cid:15) + δ ] ≤ − H x min + log 1 δ ≤ − n (cid:26) H (cid:16) αn (cid:17) , H (cid:18) α + kn (cid:19)(cid:27) + t δ + O (log n ) . (17)10hen x (cid:54) = x (cid:48) , it is easy to see that ρ x,x (cid:48) avg = Tr n − t [ U ( | x (cid:105)(cid:104) x (cid:48) | ⊗ | ψ α (cid:105)(cid:104) ψ α | ) U † ] = 0. Since the partialdecoupling theorem only applies to subnormalized states, we need to write | x (cid:105)(cid:104) x (cid:48) | using the followingrelation | x (cid:105)(cid:104) x (cid:48) | = 12 | µ + x,x (cid:48) (cid:105)(cid:104) µ + x,x (cid:48) | − | µ − x,x (cid:48) (cid:105)(cid:104) µ − x,x (cid:48) | + i | ν + x,x (cid:48) (cid:105)(cid:104) ν + x,x (cid:48) | − i | ν − x,x (cid:48) (cid:105)(cid:104) ν − x,x (cid:48) | , where | µ ± x,x (cid:48) (cid:105) = 1 √ | x (cid:105) ± | x (cid:48) (cid:105) ) , | ν ± x,x (cid:48) (cid:105) = 1 √ | x (cid:105) ± i | x (cid:48) (cid:105) ) . Then we can apply the partial decoupling theorem to the states | µ ± x,x (cid:48) (cid:105) and | ν ± x,x (cid:48) (cid:105) , and have E U (cid:107) ρ x,x (cid:48) (cid:107) ≤ (cid:18) − H x,x (cid:48) ,µ +min + 2 − H x,x (cid:48) ,µ − min + 2 − H x,x (cid:48) ,ν +min + 2 − H x,x (cid:48) ,ν − min (cid:19) , where H x,x (cid:48) ,µ ± min and H x,x (cid:48) ,ν ± min are H min ( A ∗ | RE ) Λ(Ψ , T ) with the initial state set to be | µ ± x,x (cid:48) (cid:105) and | ν ± x,x (cid:48) (cid:105) respectively. From Appendix C we know that H x,x (cid:48) min ≥ n min (cid:26) H (cid:16) αn (cid:17) , H (cid:18) α + kn (cid:19)(cid:27) − t + O (log n ) , where H x,x (cid:48) min could be any one among H x,x (cid:48) ,µ ± min and H x,x (cid:48) ,ν ± min . This gives the boundlog Pr[ (cid:107) ρ x,x (cid:48) (cid:107) ≥ δ (cid:48) ] ≤ − n (cid:26) H (cid:16) αn (cid:17) , H (cid:18) α + kn (cid:19)(cid:27) + t + log 1 δ (cid:48) + O (log n ) . (18)Now we can apply union bound and take sum of Eq. (17) over all x and Eq. (18) over all x and x (cid:48) .This means that P ( ρ x,x , ζ ) ≤ (cid:15) + δ and (cid:107) ρ x,x (cid:48) (cid:107) ≤ δ (cid:48) are satisfied for all x and x (cid:48) with probabilityat least 1 − p − p with p and p defined in Eq. (16). By Lemma 1, the code satisfies (cid:15) worst ≤ (cid:15) + δ + 2 k √ δ (cid:48) . Theorem 2.
In the large n limit, when k and t satisfy kt = o ( n ) and α = an with < a < being a constant, the expected worst-case error of the ( n, k, α ) -random code satisfies E (cid:15) worst ≤ k √ t n (cid:112) a (1 − a ) (cid:18) O (cid:18) kt n (cid:19)(cid:19) . (19) Furthermore, the probability that the worst-case error of a ( n, k, α ) -code (with respect to H × ) violatesthe inequality above is exponentially small, i.e. Pr (cid:34) (cid:15) worst > k √ t n (cid:112) a (1 − a ) (cid:18) O (cid:18) kt n (cid:19)(cid:19)(cid:35) = e − Ω( n ) . (20) Proof.
In Appendix D, we can show that P ( ρ x,x avg , ζ ) is upper bounded by the right hand side ofEq. (19). Now we apply Lemma 2 with properly chosen exponentially small δ and δ (cid:48) ,so that p and p are also exponentially small, which shows that the code satisfies (cid:15) worst = O ( n − ) withexponentially small failure probability. Since (cid:15) worst is at most 1, this exponentially probability ofviolation implies that the expectation of (cid:15) worst satisfies the inequality as well.11 .3 Remarks on noise and charge We have shown in Thm. 1 and Thm. 2 that when α = an , 0 < a <
1, the ( n, k, α )-random codesatisfies (cid:15)
Choi ≤ √ tk n (cid:112) a (1 − a ) (cid:18) O (cid:18) k t n (cid:19)(cid:19) , (cid:15) worst ≤ k √ t n (cid:112) a (1 − a ) (cid:18) O (cid:18) kt n (cid:19)(cid:19) , with high probability against erasure of t qubits, where the probability of failure (greater error) isexponentially small in n . Note that although the above analyses are done with respect to the erasureof t specific qubits, the bounds hold for the more general case of the erasure of any combination of t qubits with some probability. To show this, first note that the purified distance is independent ofthe choice of t qubits, because the permutation of the qubits commutes with the Hamming weightoperator and thus could be absorbed into the Haar measure. Then a union bound could be usedover all choices of t qubits, which at most amplifies the failure probability by a factor of (cid:0) nt (cid:1) . Since t = o ( n ) is obviously needed for the bounds to give meaningful result, the failure probability is stillexponentially small as log (cid:0) nt (cid:1) = o ( n ).Instead of erasing t out of the n qubits, a natural and stronger model of erasure noise is to haveeach qubit erased with some independent probability. Our analysis can be easily applied to thismodel by combining our bounds with the distribution on the number of qubits erased.It is also interesting to note that the code performance depends on the charge of the input ancillastate in our construction. In particular, a = 1 / α = n/
2) gives rise to the best accuracy,and the accuracy becomes worse as one increases or decreases a . The intuition is that the coderesides in a subspace with Hamming weight between α and α + k . When α = n/ k = o ( n )), the full Hilbert space with Hamming weight between α and α + k is the largest andapparently the most entangled, which enhances the performance of the code.How does our approach apply to the case without charge conservation laws? There, our constructionis modified by replacing the charge-conserving Haar-random unitary by a fully Haar random onein SU (2 n ). As long as the quantum singleton bound n − k ≥ t is satisfied, the code obtainednormally has an error rate exponentially small in n , in contrast to polynomial small for the casewith charge conservation. To be more explicit, let ∆ = n − k − t , then the randomly generatedcode will have expected Choi error e − Ω(∆) . This could be shown using an analysis similar to thecase with symmetry constraint, and the main difference is that the term relevant to the averagestate (i.e. the second term in Eq. (11)) is naturally zero in this case and the error solely comes fromthe decoupling bound. This comparison also gives an intuition for the Eastin-Knill theorem andthe lower bounds for covariant codes from a mathematical perspective.Another interesting thing to note is the dependence of error scaling on charge scaling as well asthe choice of distance measure. Here we discuss the Choi case. As we have shown, the firstterm of Eq. (11) is exponentially small in n for linear α . When α is constant, following thecalculation in Appendix A the relative entropy is at least α log n , so the second term is dominantand determines the overall distance. Now the second term with ζ set as Tr R Tr n − t Φ avg behavessignificantly differently in 1-norm and purified distance for constant charge. According to ournumerical results as illustrated in Fig. 1, the purified distance scales worse for constant α than forlinear α . To be more precise, by a linear fitting shown in Table 1 it can be seen that the error12 .0 6.5 7.0 7.5 - - - - - - - Logo f e rr o r ( pu r i f i edd i s t an c e ) α = α = α = n / α = n / - - - - - - Logo f e rr o r ( t r a c ed i s t an c e ) α = α = α = n / α = n / Figure 1: Log-log plot of the Choi error of ( n, k, α )-random code as measured by 1-norm distanceand purified distance, given different α ; here we set k = t = 2. α n/ n/ n − / when α = O (1) and n − when α = O ( n ).On the other hand, the error measured by trace distance always scales like n − . The two casesmatch the two extremes in Eq. (2). These numerical results are consistent with our calculation inAppendix B. Now let us compare the performance of our ( n, k, α )-codes with the fundamental limits of covariantcodes. For simplicity, consider the t = 1 case, namely when one qubit is erased. For U (1) symmetry,the Thm. 1 of [11] indicate the following lower bounds: (cid:15) Choi ≥ (cid:0) k (cid:100) k/ (cid:101) (cid:1) (cid:100) k/ (cid:101) k n , (cid:15) worst ≥ k n . (21)If k is large, the bound on (cid:15) Choi approaches (cid:15)
Choi ≥ n (cid:114) k π . In comparison, from Thm. 1 and Thm. 2, our ( n, k, α )-random code has smallest error when a = 1 / (cid:15) Choi ≤ √ k n (1 + O ( k /n )) , (cid:15) worst ≤ k n (1 + O ( k/n )) , so up to leading order, the worst-case distance exactly matches the lower bound, and the Choidistance matches the bound up to a constant. When k = 1, the Choi purified distance also matchesthe bound in Eq. (21) exactly. 13he situation for the general t > T A = (cid:88) α T α , where T α has support on a set of qubits α , and that α gets erased with probability q α , then (cid:15) Choi ≥ (cid:107) T L − µ ( T L ) I L (cid:107) /d L max α (∆ T α /q α ) , (cid:15) worst ≥ ∆ T L α (∆ T α /q α ) , (22)where ∆ T α and ∆ T L are the difference between the largest and smallest eigenvalues of T α and T L respectively, and µ ( T L ) is the median of the eigenvalues of T L . For example, consider the followingtwo ways to model the erasure of t qubits:1. The qubits are grouped into sets of size t , and there are n/t such sets. Each T α is the Hammingweight operator on this set, and ∆ T α = t, q α = t/n ;2. α represent all possible sets of t qubits. q α = (cid:0) nt (cid:1) − , and each T α is nt × (cid:0) nt (cid:1) − the Hammingweight operator on the set, where the coefficient is chosen so that T α sum up to the physicalcharge operate T A . In this case ∆ T α = n × (cid:0) nt (cid:1) − , q α = (cid:0) nt (cid:1) − .In either case, it turns out that max α (∆ T α /q α ) = n. As a result, the lower bounds Eq. (22) do not scale with t and are expected to be loose. We leavepotential improvement of the lower bounds in [11] as well as more careful investigation into differentmethods in e.g. [12, 19, 17, 18] for the t > In the above, we considered Haar random unitaries with charge conservation, for which one impor-tant motivation is to understand the typical performance of all such unitaries. A natural follow-upquestion of both practical and mathematical interest is how well the results hold for more “efficient”versions of random unitaries such as t -designs (“pseudorandom” distributions that match the Haarmeasure up to t moments) and random quantum circuits (circuits composed of random local gates).Consider the case without symmetries—it is known that the decoupling and error correction prop-erties of Haar random unitaries hold for (approximate) 2-designs [32], and that random circuitsconverge to t -designs in depth polynomial in t and n [33, 34], which imply that random circuits canprovide rather efficient constructions of good codes (see also [35, 36]). Do similar conclusions holdfor the case with charge conservation?In our analysis, the Haar randomness has only been used in the partial decoupling theorem, andas was noted in [27], symmetric 2-designs are sufficient for the partial decoupling bounds to hold.Therefore, all our bounds hold for 2-designs with charge conservation. Although 2-designs for14he full unitary group SU (2 n ) have been widely studied [33, 34, 37, 38], little is known about 2-designs (let alone higher order designs) with symmetry constraints, as is in our case. Especially forthe fundamental problem of convergence of random circuits to designs, we note a few interestingdifferences. Repeated applications of 2-qubit charge-conserving random unitary gates may convergeto 2-designs that we need, as was in [33, 34] for the no-symmetry case, but the proof techniques theremight be difficult to be adapted to this problem – negative values could appear in the operator basis,making the Markov chain analysis difficult; The conversion into Hamiltonian spectrum problem in[34] does not work here either, due to the complicated structure of the eigenspaces of Hammingweight operator. To summarize, there seem to be nontrivial obstacles to adapting previous proofsof convergence of random circuits to the case with charge conservation, and it remains open how toefficiently construct 2-designs with charge conservation. On the other hand, it is recently shownthat [39] in the presence of continuous symmetries like U (1), the group of unitaries generated bylocal symmetric gates is a proper subgroup of the group of global symmetric ones. As a result, undercharge conservation, local random circuits cannot converge to the Haar measure, and it remainsto be further studied whether they converge to certain t -designs. For the weaker error correctionproperty, we conjecture that random circuits composed of charge-conserving local gates are able toapproach the near-optimal performance of Haar random unitaries derived in Thm. 1 and Thm. 2with an efficiency similar to the no-symmetry case, that is, (cid:101) O ( n ) gates or O (polylog( n )) depth (forcircuit architecture without geometries). In this work we looked into U (1) covariant codes generated by Haar random unitaries, and provedthat with overwhelming probability, the error rate of such codes as measured by Choi and worst-casepurified distances scales as O ( n − ) and exactly saturates the lower bounds given in [11] at leadingorder in standard cases, and thus such random codes almost always have the best performanceallowed in the presence of certain conservation laws.We expect our results and methods to find connections or implications to several important prob-lems in physics as hinted in the introduction, as it is often crucial to take charge or energy conser-vation laws into account in physical scenarios. We shall leave detailed explorations for future work,but below we make preliminary remarks on the possible relevance to a few specific directions andpoint out some references: • Black hole information problem and Hayden–Preskill thought experiment. The Hayden–Preskill thought experiment [21] is a model of quantum information retrieval from blackhole radiation based on scrambling dynamics modeled by e.g. random circuits, which hasstimulated many key developments in our understanding of black hole information problemand quantum gravity. The basic correspondences to our error correction setup is as follows:The logical system corresponds to the quantum message thrown into the black hole, andthe ancilla state corresponds to a pure state black hole (that has not evaporated) with fixedcharge. Now the black hole dynamics subject to charge or energy conservation is modeled byour U (1)-covariant random code, the error of which against erasing the remaining black holecharacterizes how well the message can be recovered. Note that our current analysis, especially15he optimality arguments, are mostly for the regime of relatively small t (size of erasure), soto understand the connections to Hayden–Preskill it could be important to further look intothe large t regime. It is also discussed in [40] that the performance of Hayden–Preskill withcharge conservation has possible conceptual connections to the weak gravity conjecture [41]—it is possible that our analysis leads to useful quantitative statements. We refer readers toe.g. [42, 40] for more background and discussions on Hayden–Preskill with conservation laws. • Scrambling and entanglement in many-body physics. Random unitaries and circuits have alsodrawn considerable interest in condensed matter physics in recent years as “solvable” modelsof chaotic dynamics, leading to highly active fields like entanglement or operator spreading[24, 25] and measurement-induced entanglement transition [43, 44, 45]. Here conservationlaws lead to diffusive transport of the conserved quantities, and as shown in e.g. [46], the lawsof scrambling (such as operator spreading) could be fundamentally different from the casewithout conservation laws. On the other hand, the measurement-induced phase transition inrandom circuits can be understood from a quantum error correction perspective [47], indicat-ing further connections between quantum error correction and phases of matter. Given theimportance of random unitaries, symmetries and quantum error correction in these areas, wewould hope to further explore the implications of our rigorous understanding of the interplaybetween them.As mentioned, symmetries and quantum error correction are key notions in holographic quantumgravity as well. In particular, the famous conjecture about quantum gravity that it does not allowglobal symmetries is recently argued in the positive in AdS/CFT based on the quantum errorcorrection formulation of AdS/CFT [14, 15], and the argument can indeed be understood from thelimitations of covariant codes [11]. It could also be fruitful to further explore the applications ofcovariant code results and techniques there.Another future direction is to consider more general symmetries, especially SU ( d ), given its link tofault-tolerant quantum computing [13, 11]. It seems possible to generalize our method to SU ( d ),as the Hilbert space of n qudits takes the “direct-sum-product” structure under the representation U → U ⊗ n , so the partial decoupling theorem still applies. However, it remains to be calculated indetail how good the performance of the random code is.With our work, we hope to stimulate further study into random unitaries and circuits with sym-metries, which, as discussed earlier, exhibit many interesting distinctions from the no-symmetrycase. In this work we considered random global unitaries, but it would be important to furtherstudy random circuits since they can capture the “complexity” of the construction as well as thelocality structure that is important in physical scenarios. As a general program, it would be inter-esting to better understand the relations between Haar random unitaries, t -designs, and randomlocal circuits (with various architectures), in the presence of U (1) or other symmetries, throughvarious kinds of physical properties and measures. Here we take a first step by analyzing the er-ror correction performance of random unitaries with charge conservation, and conjectured that itholds for low-depth random circuits. As discussed there are difficulties in fully understanding theconvergence of symmetric random circuits to designs, but it could already be interesting and usefulto look into the behaviors of “measures” of scrambling and randomness, such as frame potentials[48, 49, 50], out-of-time-order correlators [22, 50], R´enyi entanglement entropies [22, 51, 52], which16re widely used in physics and quantum information. Acknowledgements
We thank Daniel Gottesman, Aram Harrow, Sirui Lu, Beni Yoshida, Sisi Zhou for useful discussionsand feedback. LK is supported by the ARO grant Contract Number W911NF-12-0486. ZWLis supported by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute issupported in part by the Government of Canada through the Department of Innovation, Science andEconomic Development Canada and by the Province of Ontario through the Ministry of Collegesand Universities.
A Conditional min-entropy bounds for Choi error
We first restate the partial decoupling theorem adapted to our structure of the space, which cor-responds to l j = 1 and r j = (cid:0) nj (cid:1) for all 0 ≤ j ≤ n in [27]. We denote by Π j the projector into thesubspace with Hamming weight j . Lemma 3 (Partial Decoupling) . Let T A → E be any channel mapping system A to system E , andlet Ψ AR be any joint state of system A and R . We have E U ∼ H x (cid:104)(cid:13)(cid:13) T A → E ◦ U A (cid:0) Ψ AR (cid:1) − T A → E (cid:0) Ψ AR avg (cid:1)(cid:13)(cid:13) (cid:105) ≤ − H min ( A ∗ | RE ) Λ(Ψ , T ) , where Ψ a vg AR = E U ∼ H × U Ψ AR U † . The state
Λ(Ψ , T ) is defined as Λ(Ψ , T ) = F (Ψ AR ⊗ τ ¯ AE ) F † where τ ¯ AE is the Choi-Jamiolkowski state of T and the operator F A ¯ A → A ∗ is F A ¯ A → A ∗ := n (cid:77) j =0 (cid:115) n (cid:0) nj (cid:1) (cid:16) Π Aj ⊗ Π ¯ Aj (cid:17) . For simplicity we define the 2 m -qubit normalized state | φ ( m ) i (cid:105) = (cid:18) mi (cid:19) − / (cid:88) v ∈{ , } m , | v | = i | v (cid:105)| v (cid:105) , which is the maximally entangled state between two copies of subspaces with Hamming weight i of m qubits. We also define | ˆ φ ( m ) (cid:105) as m EPR pairs.In our setting the state Ψ AR is the encoded state before applying the random unitary, which is k EPR pairs appended by the fixed state | ψ (cid:105) ,Ψ AR = | Ψ AR (cid:105)(cid:104) Ψ AR | , | Ψ AR (cid:105) = | ˆ φ (cid:105) A R ⊗ | ψ α (cid:105) A . and A refers to different parts of A , and have k and n − k qubits each. Then it is easy to seethat Π Aj | Ψ (cid:105) AR = (cid:115) (cid:0) kj − α (cid:1) k | φ ( k ) j − α (cid:105) A R | ψ α (cid:105) A . The channel T traces over n − t qubits, so the corresponding Choi-Jamiolkowski is τ ¯ AE = Tr n − t | ˆ φ ( n ) (cid:105)(cid:104) ˆ φ ( n ) | = | ˆ φ ( t ) (cid:105)(cid:104) ˆ φ ( t ) | ¯ A E ⊗ I ¯ A n − t . Let Π ( a ) b be the subspace on a qubits with Hamming weight b . We haveΠ ¯ Aj τ ¯ AE Π ¯ Aj (cid:48) = 12 n − t (cid:88) i Π ¯ Aj (cid:104) | ˆ φ ( t ) (cid:105)(cid:104) ˆ φ ( t ) | ¯ A E (Π ( n − t ) i ) ¯ A (cid:105) Π ¯ Aj (cid:48) = 12 n (cid:88) i (cid:115)(cid:18) tj − i (cid:19)(cid:18) tj (cid:48) − i (cid:19) | φ ( t ) j − i (cid:105)(cid:104) φ ( t ) j (cid:48) − i | ¯ A E (Π ( n − t ) i ) ¯ A , (23)and thereforeΛ(Ψ , T ) = 12 k (cid:88) i,j,j (cid:48) (cid:118)(cid:117)(cid:117)(cid:116) (cid:0) tj − i (cid:1)(cid:0) tj (cid:48) − i (cid:1)(cid:0) kj − α (cid:1)(cid:0) kj (cid:48) − α (cid:1)(cid:0) nj (cid:1)(cid:0) nj (cid:48) (cid:1) | φ ( t ) j − i (cid:105)(cid:104) φ ( t ) j (cid:48) − i | ¯ A E (Π ( n − t ) i ) ¯ A | φ ( k ) j − α (cid:105)(cid:104) φ ( k ) j (cid:48) − α | A R | ψ α (cid:105)(cid:104) ψ α | A = (cid:88) i,j,j (cid:48) (Π ( n − t ) i ) ¯ A ⊗ | γ j,i (cid:105)(cid:104) γ j (cid:48) ,i | ¯ A A A ER = (cid:88) i (Π ( n − t ) i ) ¯ A ⊗ | Γ i (cid:105)(cid:104) Γ i | ¯ A A A ER where | Γ i (cid:105) = (cid:88) j | γ j,i (cid:105) , | γ j,i (cid:105) ¯ A A A ER = (cid:118)(cid:117)(cid:117)(cid:116) (cid:0) tj − i (cid:1)(cid:0) kj − α (cid:1) k (cid:0) nj (cid:1) | φ ( t ) j − i (cid:105) ¯ A E | φ ( k ) j − α (cid:105) A R | ψ α (cid:105) A . As mentioned in Section 2.3, the min conditional entropy is defined as H min ( P | Q ) ρ = sup σ ≥ , Tr σ =1 sup { λ ∈ R | − λ I P ⊗ σ Q ≥ ρ P Q } , or equivalently, H min ( P | Q ) ρ = − log s where s is the optimum value of the following SDP s = inf Tr σ, s.t. I P ⊗ σ Q ≥ ρ P Q , σ ≥ . (24)The corresponding dual SDP is t = sup (cid:104) ρ P Q , y
P Q (cid:105) , s.t. Tr P [ y P Q ] ≤ I Q , y ≥ . (25)It is obvious that both primal and dual are strongly feasible, so s = t . We can use the followinglemma to relate the min-entropy of Λ(Ψ , T ) to the min-entropy of each | Γ i (cid:105) .18 emma 4. Suppose the register P in eqs. (24)(25) can be divided into P and P , and the state ρ P Q has the structure ρ P Q = m (cid:88) i =1 Π P i ⊗ ρ P Qi where Π i are projectors into disjoint subspaces. Then s , the result of the SDP, satisfy m (cid:88) i s i ≤ s ≤ (cid:88) i s i where s i is the result for the SDP of ρ i .Proof. We prove the lemma by constructing feasible solutions of the primal and the dual. Let σ = (cid:88) i σ i where σ i is the optimal solution for the primal SDP of ρ i . Then it is natural that Tr[ σ ] = (cid:80) i s i ,and the condition holds because I P P ⊗ σ ≥ (cid:88) i Π P i ⊗ I P ⊗ σ Qi ≥ (cid:88) i Π P i ⊗ ρ P Qi = ρ. For the dual SDP, let y = 1 m (cid:88) i Π P i Tr[Π i ] ⊗ y P Qi where y i is the optimal solution for the dual SDP of ρ i . Then (cid:104) ρ, y (cid:105) = 1 m (cid:88) i Tr (cid:20) Π i Π i Tr[Π i ] (cid:21) Tr[ y i ρ i ] = 1 m (cid:88) i s i , and Tr P y = 1 m (cid:88) i Tr Π i Tr[Π i ] Tr P [ y i ] ≤ m (cid:88) i I Q = I Q . Note that the state | Γ i (cid:105) is a pure state, so its min entropy can be calculated using Eq. (7). Thereforewe have H min ( A ∗ | RE ) Γ i = − (cid:88) j (cid:113) k (cid:0) nj (cid:1) (cid:18) tj − i (cid:19)(cid:18) kj − α (cid:19) , and the value for the corresponding SDP is (cid:88) j (cid:113) k (cid:0) nj (cid:1) (cid:18) tj − i (cid:19)(cid:18) kj − α (cid:19) . j and i should satisfy 0 ≤ j − i ≤ t, ≤ j − α ≤ k, so α − t ≤ i ≤ α + k , and there are at most k + t + 1 possible values for i . By Lemma 4 we have − log κ ≤ H min ( A ∗ | RE ) Λ ≤ − log κk + t + 1 . (26)where κ = (cid:88) i (cid:88) j (cid:113) k (cid:0) nj (cid:1) (cid:18) tj − i (cid:19)(cid:18) kj − α (cid:19) . Note that 1 (cid:113) k (cid:0) nj (cid:1) (cid:18) tj − i (cid:19)(cid:18) kj − α (cid:19) ≤ − k/ (cid:18) tt/ (cid:19)(cid:18) kk/ (cid:19) (cid:113) min { (cid:0) nα (cid:1) , (cid:0) nα + k (cid:1) } , so from Eq. (26) we have H min ( A ∗ | RE ) Λ ≥ n min (cid:26) H (cid:16) αn (cid:17) , H (cid:18) α + kn (cid:19)(cid:27) − t − k + O (log n )for general values of t and k as long as α is linear in n . Here H ( · ) is the binomial entropy function H ( x ) = − x log x − (1 − x ) log(1 − x ) , ≤ x ≤ . If t k = O ( n ), this implies H min ( A ∗ | RE ) Λ = Ω( n ).When α , k and t are all O (1) and does not depend on n , the bound in Eq. (26) imples that H min ( A ∗ | RE ) Λ ≥ α log n + O (1) . B Average state and Choi error
Following the previous definitions, Φ avg is a joint state on n -qubit register A and k -qubit register R . From Eq. (8), Φ RA avg = E U ∼ H × U ( | ˆ φ ( k ) (cid:105)(cid:104) ˆ φ ( k ) | A R ⊗ | ψ α (cid:105)(cid:104) ψ α | A ) U † = k (cid:88) j =0 Tr A [Π Aj ( | ˆ φ ( k ) (cid:105)(cid:104) ˆ φ ( k ) | A R ⊗ | ψ α (cid:105)(cid:104) ψ α | A )Π Aj ] ⊗ Π Aj (cid:0) nj (cid:1) =2 − k k + α (cid:88) j = α Π Rj − α ⊗ Π Aj (cid:0) nj (cid:1) , n − t Φ RA avg = 2 − k k + α (cid:88) j = α t (cid:88) i =0 Π Rj − α ⊗ Π Ei (cid:0) n − tj − i (cid:1)(cid:0) nj (cid:1) = 2 − k k (cid:88) j =0 t (cid:88) i =0 Π Rj ⊗ Π Ei (cid:0) n − tj + α − i (cid:1)(cid:0) nj + α (cid:1) , where E refers to the t -qubit register that the complementary channel maps to. To get an upperbound for the second term of Eq. (11), we can replace the minimization over ζ by an arbitrary fixed ζ , which we choose to be the marginal state ζ E = Tr R Tr n − t Φ RA avg = 2 − k t (cid:88) i =0 Π Ei k + α (cid:88) j = α (cid:0) kj − α (cid:1)(cid:0) n − tj − i (cid:1)(cid:0) nj (cid:1) . We define β i = 2 − k k + α (cid:88) j = α (cid:0) kj − α (cid:1)(cid:0) n − tj − i (cid:1)(cid:0) nj (cid:1) = 2 − k k (cid:88) j =0 (cid:0) kj (cid:1)(cid:0) n − tj + α − i (cid:1)(cid:0) nj + α (cid:1) , (27)so that ζ E = t (cid:88) i =0 β i Π Ei Note that all the states are diagonal, the fidelity is given by F (cid:18) Tr n − t Φ avg , I k ⊗ ζ (cid:19) = k (cid:88) j =0 t (cid:88) i =0 Tr[Π Rj ⊗ Π Ai ] (cid:118)(cid:117)(cid:117)(cid:116) − k β i × − k (cid:0) n − tj + α − i (cid:1)(cid:0) nj + α (cid:1) =2 − k k (cid:88) j =0 t (cid:88) i =0 (cid:18) kj (cid:19)(cid:18) ti (cid:19)(cid:118)(cid:117)(cid:117)(cid:116) β i (cid:0) n − tj + α − i (cid:1)(cid:0) nj + α (cid:1) (28)For any nonnegative number n and real number x , we define x n = x ( x − . . . ( x − n + 1) , x n = x ( x + 1) . . . ( x + n − . They are related by x n = ( x + n − n , x n = ( x − n + 1) n and x n = ( − x ) n ( − n , x n = ( − x ) n ( − n . It could be verified that the following binomial theorems hold (by induction on n )( x + y ) n = n (cid:88) k =0 (cid:18) nk (cid:19) x k y n − k , ( x + y ) n = n (cid:88) k =0 (cid:18) nk (cid:19) x k y n − k . β i in Eq. (27) we have β i =2 − k k (cid:88) j =0 (cid:0) kj (cid:1)(cid:0) n − tj + α − i (cid:1)(cid:0) nj + α (cid:1) = 1 n t k k (cid:88) j =0 (cid:18) kj (cid:19) ( j + α ) i ( n − α − j ) t − i = 1 n t k (cid:88) j,x (cid:18) kj (cid:19)(cid:18) ix (cid:19) j x α i − x ( − t − i ( j − n + α ) t − i = 1 n t k (cid:88) j,x (cid:18) kj (cid:19)(cid:18) ix (cid:19) j x α i − x ( − t − i ( j − n + α + t − i − t − i = 1 n t k (cid:88) j,x,y (cid:18) kj (cid:19)(cid:18) ix (cid:19) j x α i − x ( − t − i (cid:18) t − iy (cid:19) ( j − x ) y ( x − n + α + t − i − t − i − y = 1 n t k (cid:88) j,x,y (cid:18) kj (cid:19) j x + y (cid:18) ix (cid:19) α i − x ( − y (cid:18) t − iy (cid:19) ( − x + n − α − t + i + 1) t − i − y = 1 n t (cid:88) x,y − ( x + y ) ( − y k x + y (cid:18) ix (cid:19)(cid:18) t − iy (cid:19) α i − x ( n − α − x − y ) t − i − y . Suppose that kt = o ( n ), we can see that the term with x = x and y = y is of order ( kt/n ) x + y times the term with x = y = 0. Then by keeping the terms x + y ≤ β i = α i ( n − α ) t − i n t (cid:18) ik − akt (2 a − a ) n + ξ a − a n + O (cid:18) k t n (cid:19)(cid:19) , where a = α/n and ξ = k (cid:0) a ( k − t − t + i ( − a + k + 3) + i (2 a (2 a ( t − − ( k + 1) t + k + 3) − k − (cid:1) . Similarly (cid:0) n − tj + α − i (cid:1)(cid:0) nj + α (cid:1) = α i ( n − α ) t − i n t (cid:18) j ( at − i )( a − an + ξ a − a n + O (cid:18) k t n (cid:19)(cid:19) with ξ = j (cid:0) a ( j − t − t + i ( − a + j + 1) + i (2 a ( a ( t −
1) + j ( − t ) + j + 1) − j − (cid:1) . Therefore, (cid:118)(cid:117)(cid:117)(cid:116) β i (cid:0) n − tj + α − i (cid:1)(cid:0) nj + α (cid:1) = α i ( n − α ) t − i n t (cid:18) j + k )( at − i )4( a − an + ξ a − a n + O (cid:18) k t n (cid:19)(cid:19) , ξ = a t (cid:0) j ( t −
2) + 4 j (( k − t + 2) + k ( k ( t − − t + 2) (cid:1) + i (cid:0) j ( − a + k + 2) + k ( − a + k + 6) + 4 j (cid:1) + 2 i (cid:0) − j ( a ( t −
2) + 1) + j (4 a (2 a ( t − − kt + 2) − (cid:1) − ik (( a ( − a ( t −
1) + ( k + 2) t − k + 3)) + k + 3)) . Then we can multiply this by (cid:0) ti (cid:1) and sum over i to have the fidelity (cid:88) i (cid:18) ti (cid:19)(cid:118)(cid:117)(cid:117)(cid:116) β i (cid:0) n − tj + α − i (cid:1)(cid:0) nj + α (cid:1) = 1 − t ( k − j ) a (1 − a ) n + O (cid:18) k t n (cid:19) . (29)Now we plug this into Eq. (28) and have F (cid:18) Tr n − t Φ avg , I k ⊗ ζ (cid:19) = 1 − tk a (1 − a ) n + O (cid:18) k t n (cid:19) , and the corresponding purified distance is P (cid:18) Tr n − t Φ avg , I k ⊗ ζ (cid:19) = √ tk n (cid:112) a (1 − a ) (cid:18) O (cid:18) t k n (cid:19)(cid:19) . Another interesring case to consider is α = O (1), which could be directly evaluated from Eq. (28)for small values of k and t . For example, when k = t = 1, we have F (cid:18) Tr n − t Φ avg , I k ⊗ ζ (cid:19) = 1 + −√ − √ α + (cid:112) α (2 α + 1) + (cid:112) ( α + 1)(2 α + 1)2 √ n + O ( n − ) , and when k = t = 2, F (cid:18) Tr n − t Φ avg , I k ⊗ ζ (cid:19) = 1 + − − α + (cid:112) α ( α + 1) + (cid:112) ( α + 1)( α + 2)2 n + 1 + O ( n − ) . In both cases the purified distance is O ( n − / ). C Conditional min-entropy bounds for worst-case error
We are interested in the conditional entropy H min ( A ∗ | RE ) Λ(Ψ , T ) with the initial states Ψ being | x (cid:105) , | µ ± x,x (cid:48) (cid:105) and | ν ± x,x (cid:48) (cid:105) , where | µ ± x,x (cid:48) (cid:105) = 1 √ | x (cid:105) ± | x (cid:48) (cid:105) ) , | ν ± x,x (cid:48) (cid:105) = 1 √ | x (cid:105) ± i | x (cid:48) (cid:105) ) .
23n other words, the state | Ψ AR (cid:105) is one of the above states appended by | ψ α (cid:105) , a state with Hammingweight α . The reference system R is now trivial, in contrast to the k qubits in Appendix A. Herethe channel T is the erasure channel over n − t qubits. Using Eq. (23), we haveΛ(Ψ , T ) = 1 (cid:0) n | x | + α (cid:1) | x (cid:105)(cid:104) x | A ⊗ | ψ α (cid:105)(cid:104) ψ α | A ⊗ (cid:88) i (cid:18) t | x | + α − i (cid:19) | φ ( t ) | x | + α − i (cid:105)(cid:104) φ ( t ) | x | + α − i | ¯ A E (Π ( n − t ) i ) ¯ A = (cid:88) i (Π ( n − t ) i ) ¯ A ⊗ | γ x,i (cid:105)(cid:104) γ x,i | , (30)where | γ x,i (cid:105) = (cid:118)(cid:117)(cid:117)(cid:116) (cid:0) t | x | + α − i (cid:1)(cid:0) n | x | + α (cid:1) | x (cid:105) A ⊗ | ψ α (cid:105) A ⊗ | φ ( t ) | x | + α − i (cid:105) ¯ A E . Now using Lemma 4 and Eq. (7), we have − log (cid:80) i (cid:0) t | x | + α − i (cid:1) (cid:0) n | x | + α (cid:1) ≤ H x min ≤ − log (cid:80) i (cid:0) t | x | + α − i (cid:1) (cid:0) n | x | + α (cid:1) ( t + 1) , where H x min stands for H min ( A ∗ | RE ) Λ(Ψ , T ) when the initial state is | x (cid:105)(cid:104) x | . This could be furthersimplified to − log (cid:34) (cid:0) tt (cid:1)(cid:0) n | x | + α (cid:1) (cid:35) ≤ H x min ≤ − log (cid:34) (cid:0) tt (cid:1)(cid:0) n | x | + α (cid:1) ( t + 1) (cid:35) . When the initial state is | µ ± x,x (cid:48) (cid:105) and | ν ± x,x (cid:48) (cid:105) , the state in Eq. (30) will have the same form, with | γ x,i (cid:105) replaced by √ ( | γ x,i (cid:105) ± | γ x (cid:48) ,i (cid:105) ) and √ ( | γ x,i (cid:105) ± i | γ x (cid:48) ,i (cid:105) ) correspondingly. Then − log (cid:104) χ (cid:105) ≤ H x,x (cid:48) min ≤ − log (cid:20) χ t + 1) (cid:21) , where H x,x (cid:48) min stands for H min ( A ∗ | RE ) Λ(Ψ , T ) when the initial state is one of | µ ± x,x (cid:48) (cid:105) or | ν ± x,x (cid:48) (cid:105) , and χ = (cid:88) i (cid:0) t | x | + α − i (cid:1)(cid:113)(cid:0) n | x | + α (cid:1) + (cid:0) t | x (cid:48) | + α − i (cid:1)(cid:113)(cid:0) n | x (cid:48) | + α (cid:1) = (cid:0) tt (cid:1)(cid:0) n | x |− α (cid:1) + (cid:0) tt (cid:1)(cid:0) n | x (cid:48) |− α (cid:1) + 2 (cid:0) tt + | x |−| x (cid:48) | (cid:1)(cid:113)(cid:0) n | x |− α (cid:1)(cid:0) n | x (cid:48) |− α (cid:1) . Suppose that in the large n limit αn and α + kn are both constants between 0 and 1, we have H x min = nH (cid:18) | x | + αn (cid:19) − t + O (log n ) ≥ n min (cid:26) H (cid:16) αn (cid:17) , H (cid:18) α + kn (cid:19)(cid:27) − t + O (log n ) , and H x,x (cid:48) min ≥ n min (cid:26) H (cid:18) | x | + αn (cid:19) , H (cid:18) | x (cid:48) | + αn (cid:19)(cid:27) − t + O (log n ) ≥ n min (cid:26) H (cid:16) αn (cid:17) , H (cid:18) α + kn (cid:19)(cid:27) − t + O (log n ) . Average state and worst-case error
It is easy to see that E U ∼ H × U ( | x (cid:105)(cid:104) x (cid:48) | ⊗ | ψ (cid:105)(cid:104) ψ | ) U † = (cid:40) , x (cid:54) = x (cid:48) Π ( n ) | x | + α / (cid:0) n | x | + α (cid:1) , x = x (cid:48) . In the case of x = x (cid:48) , we take trace over n − t qubits and have ρ x,x avg = t (cid:88) i =0 Π ( t ) i (cid:0) n − t | x | + α − i (cid:1)(cid:0) n | x | + α (cid:1) , and we wish to show that this is close to some fixed state ζ independent of x . We propose that ζ is ρ x,x avg averaged over x , ζ = 12 k k (cid:88) j =0 Π ( t ) i (cid:0) kj (cid:1)(cid:0) n − tj + α − i (cid:1)(cid:0) nj + α (cid:1) = (cid:88) i β i Π ( t ) i , where β i is the quantity previously defined in Eq. (27) from Appendix B. For any x , the fidelity isgiven by F ( ρ x,x avg , ζ ) = (cid:88) i (cid:18) ti (cid:19)(cid:118)(cid:117)(cid:117)(cid:116) β i (cid:0) n − t | x | + α − i (cid:1)(cid:0) n | x | + α (cid:1) . which is exactly the result in Eq. (29) with j = | x | . Then we havemax x P ( ρ x,x avg , ζ ) = k √ t n (cid:112) a (1 − a ) (cid:18) (cid:18) kt n (cid:19)(cid:19) . References [1] Peter W. Shor. Scheme for reducing decoherence in quantum computer memory.
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