Permutation-Invariant Quantum Codes for Deletion Errors
aa r X i v : . [ qu a n t - ph ] F e b Permutation-Invariant Quantum Codes forDeletion Errors
Taro Shibayama ∗ Manabu Hagiwara ∗ Abstract
This paper presents conditions for constructing permutation-invariantquantum codes for deletion errors and provides a method for construct-ing them. Our codes give the first example of quantum codes that cancorrect two or more deletion errors. Also, our codes give the first ex-ample of quantum codes that can correct both multiple-qubit errorsand multiple-deletion errors. We also discuss a generalization of theconstruction of our codes at the end.
Quantum error-correcting codes are one of the essential factors for the de-velopment of quantum computers, which were first introduced by Shor in1995 [22]. Since then, many codes have been constructed, e.g., CSS codes [2],stabilizer codes [4], surface codes [3], etc.Recently, in 2019, Leahy et al. provided a way to turn quantum insertion-deletion errors into errors that can be handled by conventional methods underthe specific assumption [10]. Since then, quantum deletion error-correctionhave attracted a lot of attention from researchers. In quantum informationtheory, quantum deletion error-correction is a problem of determining thequantum state in the entire quantum system from a quantum state in apartial system. Therefore it is related to various topics, e.g., quantum erasureerror-correcting codes [5], quantum secret sharing [7], purification of quantumstate [8], quantum cloud computing [1], etc.The first quantum deletion codes under the general scenario was con-structed in 2020 [11]. Since then, several quantum deletion codes have been ∗ Department of Mathematics and Informatics, Graduate School of Science, Chiba Uni-versity 1-33 Yayoi-cho, Inage-ku, Chiba City, Chiba Pref., JAPAN, 263-0022
Throughout this paper, we fix the following notation. Let N ≥ be an integerand [ N ] := { , , . . . , N } . For a square matrix M over the complex field C ,we denote by Tr( M ) the sum of the diagonal elements of M . Set | i , | i ∈ C as | i := (1 , ⊤ , | i := (0 , ⊤ , and | x i := | x i ⊗ | x i ⊗ · · · ⊗ | x N i ∈ C ⊗ N for a bit sequence x = x x · · · x N ∈ { , } N . Here ⊗ is the tensor productoperation and ⊤ is the transpose operation. Let h x | := | x i † denote theconjugate transpose of | x i . We define the Hamming weight wt( x ) of x by wt( x ) := { p ∈ [ N ] | x p = 0 } .A positive semi-definite Hermitian matrix of trace is called a densitymatrix. We denote by S ( C ⊗ N ) the set of all density matrices of order N .An element of S ( C ⊗ N ) is called a quantum state. We also use a complexvector | ψ i ∈ C ⊗ N for representing a pure quantum state | ψ ih ψ | ∈ S ( C ⊗ N ) .2et M = P x , y ∈{ , } N m x , y | x ih y |⊗· · ·⊗| x N ih y N | be a square matrix with m x , y ∈ C . For an integer p ∈ [ N ] , define a map Tr p : S ( C ⊗ N ) → S ( C ⊗ ( N − ) as Tr p ( M ) := X x , y ∈{ , } N m x , y · Tr( | x p ih y p | ) | x ih y |⊗· · · ⊗ | x p − ih y p − | ⊗ | x p +1 ih y p +1 |⊗· · · ⊗ | x N ih y N | . The map Tr p is called a partial trace. Recall that in classical coding theory, for an integer ≤ t < N , a t -deletionerror is defined as a map from a bit sequence of length N to its subsequenceof length N − t . We define a t -deletion error in the terms of quantum codingtheory. Definition 3.1 (Deletion Error D P ) . Let ≤ t < N be an integer. For a set P = { p , . . . , p t } ⊂ [ N ] with p < · · · < p t , define a map D P : S ( C ⊗ N ) → S ( C ⊗ ( N − t ) ) as D P ( ρ ) := Tr p ◦ · · · ◦ Tr p t ( ρ ) , where ρ ∈ S ( C ⊗ N ) is a quantum state. Here the symbol ◦ indicates thecomposition of maps. We call the map D P a t -deletion error with the deletionposition P . The cardinality | P | is called the number of deletions for D P . In this paper, an integer ≤ t < N is fixed and denotes a number ofdeletions, and a set P ⊂ [ N ] denotes the deletion position satisfying | P | = t . Definition 3.2 (Deletion Error-Correcting Code) . We call an image of
Enc an [ N, K ] t -deletion error-correcting code if the following conditions hold. • There exists a map
Enc : C ⊗ K → C ⊗ N defined as the compositionof two maps enc ◦ pad N,K , where the map pad
N,K : C ⊗ K → C ⊗ N is defined by pad N,K ( | ψ i ) := | ψ i ⊗ | i ⊗ · · · ⊗ | i , and the map enc : C ⊗ N → C ⊗ N is a unitary transformation acting on C ⊗ N . There exists a map
Dec : S ( C ⊗ ( N − t ) ) → C ⊗ K defined by the operationsallowed by quantum mechanics, and Dec ◦ D P ◦ Enc( | ψ i ) = | ψ i holds for any quantum state | ψ i ∈ C ⊗ K and any deletion position P ⊂ [ N ] satisfying | P | = t .In other words, there exist an encoder Enc and a decoder
Dec that correctany t -deletion errors. Note that a t -deletion error-correcting code is an s -deletion error-correctingcode for any positive integer s ≤ t .The following Definition 3.3 is the conditions for constructing our codes.Note that for binomial coefficients, if w < or N < w , we define (cid:0) Nw (cid:1) := 0 . Definition 3.3 (Conditions (D1), (D2), and (D3)) . For two non-empty sets
A, B ⊂ { , , . . . , N } and a map f : A ∪ B → C , define three conditions (D1),(D2), and (D3) as follows:(D1) X w ∈ A | f ( w ) | (cid:18) Nw (cid:19) = X w ∈ B | f ( w ) | (cid:18) Nw (cid:19) = 1 .(D2) For any integer ≤ k ≤ t , X w ∈ A | f ( w ) | (cid:18) N − tw − k (cid:19) = X w ∈ B | f ( w ) | (cid:18) N − tw − k (cid:19) = 0 . (D3) For any integers w , w ∈ A ∪ B , w = w = ⇒ | w − w | > t. The following Theorem 3.4 is the main theorem of this paper and describesa new construction method for quantum deletion error-correcting codes.
Theorem 3.4.
Let
A, B ⊂ { , } N be non-empty sets with A ∩ B = ∅ and f : A ∪ B → C be a map satisfying the conditions (D1), (D2), and (D3). Thenthe code Q fA,B is an [ N, t -deletion error-correcting code with the encoder Enc fA,B and the decoder
Dec fA,B . Here, the notations Q fA,B , Enc fA,B , and
Dec fA,B above are defined in thenext section. 4
Proof of The Main Theorem
In this section, we shall give the proof of Theorem 3.4. The proofs of Lemmasin this section are given in later appendices.
Definition 4.1 (Encoder
Enc fA,B and Code Q fA,B ) . Let
A, B ⊂ { , , . . . , N } be non-empty sets with A ∩ B = ∅ and f : A ∪ B → C be a map satisfyingthe conditions (D1), (D2), and (D3). Let us define an encoder as a linearmap Enc fA,B : C → C ⊗ N . For a quantum state | ψ i = α | i + β | i ∈ C , Enc fA,B maps the state | ψ i to the following state | Ψ i , | Ψ i := X x ∈{ , } N wt( x ) ∈ A αf (wt( x )) | x i + X y ∈{ , } N wt( y ) ∈ B βf (wt( y )) | y i . (1) Set Q fA,B as the image of Enc fA,B , i.e., Q fA,B := { Enc fA,B ( | ψ i ) | | ψ i ∈ C , | ψ ih ψ | ∈ S ( C ) } . The map
Enc fA,B can be written as a form enc ◦ pad N,K with some unitarytransformation enc , thus it satisfies the condition of Definition 3.2. Note thatfor the vector | Ψ i defined by Equation (1), h Ψ | Ψ i = X x ∈{ , } N wt( x ) ∈ A | αf (wt( x )) | + X y ∈{ , } N wt( y ) ∈ B | βf (wt( y )) | = X w ∈ A | αf ( w ) | (cid:18) Nw (cid:19) + X w ∈ B | βf ( w ) | (cid:18) Nw (cid:19) = | α | + | β | = 1 holds by the condition (D1). Therefore, | Ψ i is a quantum state.A quantum state is permutation-invariant if the state is invariant underposition permutations. A quantum code is permutation-invariant if any stateof the code is permutation-invariant. The term permutation-invariant is alsocalled as PI. The code Q fA,B is a PI code. The following Lemma 4.2 describesthe state after a deletion error for a PI state. Lemma 4.2.
Let | Ψ i be a pure PI state with | Ψ i := X x ∈{ , } N c (wt( x )) | x i , (2)5 or a map c : { , , . . . , N } → C . For an integer ≤ k ≤ t , | Ψ k i := X x ∈{ , } N − t c (wt( x ) + k ) | x i . (3) Then, for any deletion position P ⊂ [ N ] satisfying | P | = t , D P ( | Ψ ih Ψ | ) = t X k =0 (cid:18) tk (cid:19) | Ψ k ih Ψ k | . Note that Equation (1) is obtained in Equation (2) by setting c ( w ) = αf ( w ) w ∈ A,βf ( w ) w ∈ B, . (4)Equation (3) in our codes can be expressed in good form by the followingLemma 4.3. The conditions (D2) and (D3) can be considered as an adapta-tion of the Knill and Laflamme conditions [9] to PI codes for deletion errors. Lemma 4.3.
Let
A, B ⊂ { , } N be non-empty sets with A ∩ B = ∅ and f : A ∪ B → C be a map satisfying the conditions (D2) and (D3). Thenfor any integer ≤ k ≤ t , there exist a real number l k = 0 and vectors | u k i , | u k i ∈ C ⊗ ( N − t ) that satisfy the followings: • For a vector | Ψ k i defined by Equations (3) and (4), | Ψ k i = l k ( α | u k i + β | u k i ) . • For integers k , k ∈ { , , . . . , t } and b , b ∈ { , } , h u k b | u k b i = ( k , b ) = ( k , b ) , k , b ) = ( k , b ) . A set P := { P k } of projection matrices of order N is called a projectivemeasurement if and only if P k P k = I , where k is an index and I is theidentity matrix of order N . If the quantum state immediately before themeasurement is ρ ∈ S ( C ⊗ N ) then the probability that outcome k occursis given by Tr( P k ρ ) , and the quantum state ρ ′ after the measurement is ρ ′ := P k ρP k Tr( P k ρ ) . 6 efinition 4.4 (Set P A,B ) . Let
A, B ⊂ { , } N be sets. For an integer ≤ k ≤ t , suppose that W k := { x ∈ { , } N − t | wt( x ) + k ∈ A ∪ B } . Then we define a set P A,B := { P , P , . . . , P t , P ∅ } of projection matrices,where P k := P x ∈ W k | x ih x | k ∈ { , , . . . , t } , I − P tk =0 P k k = ∅ . If non-empty sets
A, B ⊂ { , } N satisfy the condition (D3), the set P A,B is clearly a projective measurement. The following Lemma 4.5 shows theresults of the projective measurement P A,B under the state after a deletionerror in our code Q fA,B . Lemma 4.5.
Let
A, B ⊂ { , } N be non-empty sets with A ∩ B = ∅ and f : A ∪ B → C be a map satisfying the conditions (D1), (D2), and (D3). Let | Ψ i and | Ψ k i for an integer ≤ k ≤ t be defined by Equations (2), (3) and(4). If we perform the projective measurement P A,B under the quantum state D P ( | Ψ ih Ψ | ) ∈ S ( C ⊗ ( N − t ) ) for any deletion position P ⊂ [ N ] , the probability p ( k ) of getting outcome k ∈ { , , . . . , t, ∅ } is p ( k ) = ( (cid:0) tk (cid:1) l k k ∈ { , , . . . , t } , k = ∅ . When the outcome k ∈ { , , . . . , t } is obtained, the quantum state ρ ( k ) ∈ S ( C ⊗ ( N − t ) ) after the measurement is ρ ( k ) = 1 l k | Ψ k ih Ψ k | . Definition 4.6 (Error-Correcting Operator U k ) . Suppose the assumptions ofLemma 4.3 are satisfied. Then, for integers k ∈ { , , . . . , t } and m ∈ { , } ,we can choose a unitary matrix U k whose m th row is h u km | . We call the matrix U k an error-correcting operator. The m th row of U k for ≤ m ≤ N − t isalso denoted as h u km | . Definition 4.7 (Decoding Algorithm
Dec fA,B ) . Let
A, B ⊂ { , } N be non-empty sets with A ∩ B = ∅ and f : A ∪ B → C be a map satisfying theconditions (D1), (D2), and (D3). Define a decoder Dec fA,B as a map from ρ ∈ S ( C ⊗ ( N − t ) ) to σ ∈ C constructed by the following steps: Step 1) Perform the projective measurement P A,B under the quantum state ρ . Assume that the outcome is k and that the state after the mea-surement is ρ ( k ) .(Step 2) Let ˜ ρ := U k ρ ( k ) U † k . Here U k is the error-correcting operator.(Step 3) At last, return σ := Tr ◦ · · · ◦ Tr | {z } ( N − t − times ( ˜ ρ ) . We now describe the proof of the main theorem.
Proof of Theorem 3.4.
Set | Ψ i := Enc fA,B ( | ψ i ) for a pure quantum state | ψ i ∈ C . For an integer k ∈ { , , . . . , t } and integers i, j ∈ [2 N − t ] , wedenote the ( i, j ) element of the matrix U k (cid:16) l k | Ψ k ih Ψ k | (cid:17) U † k by u k ( i, j ) . ByLemma 4.3, we have u k ( i, j ) = h u ki | (cid:18) l k | Ψ k ih Ψ k | (cid:19) | u kj i = h u ki | ( α | u k i + β | u k i )( α h u k | + β h u k | ) | u kj i = ( α h u ki | u k i + β h u ki | u k i )( α h u k | u kj i + β h u k | u kj i )= | α | ( i, j ) = (1 , ,αβ ( i, j ) = (1 , ,αβ ( i, j ) = (2 , , | β | ( i, j ) = (2 , , . (5)By Lemmas 4.2, 4.5, Definition 4.7, and Equation (5), Dec fA,B ◦ D P ◦ Enc fA,B ( | ψ ih ψ | )= Dec fA,B ◦ D P ( | Ψ ih Ψ | )= Dec fA,B t X k =0 (cid:18) tk (cid:19) | Ψ k ih Ψ k | ! = Tr ◦ · · · ◦ Tr (cid:18) U k (cid:18) l k | Ψ k ih Ψ k | (cid:19) U † k (cid:19) = Tr ◦ · · · ◦ Tr ( | ih | ⊗ · · · ⊗ | ih | ⊗ | ψ ih ψ | )= | ψ ih ψ | holds for any pure quantum state | ψ i ∈ C and any deletion position P ∈ [ N ] .This is exactly the original quantum state.8 Examples
By Theorem 3.4, we can construct a permutation-invariant quantum code fordeletion errors from finding two non-empty sets
A, B ⊂ { , , . . . , N } with A ∩ B = ∅ and a map f : A ∪ B → C that satisfy the three conditions (D1),(D2), and (D3). we give two families of our codes in this section. First, we introduce a key combinatorial equation in giving the first example.
Lemma 5.1.
Let n be a positive integer. Then for all integers ≤ t ≤ n − , n X l =0 (cid:18) nl (cid:19) l t ( − l = 0 . Lemma 5.1 can be easily shown by induction using the binomial identity P nl =0 (cid:0) nl (cid:1)(cid:0) lt (cid:1) ( − l = 0 , which holds for any integer ≤ t < n .The following Theorem 5.2 gives quantum codes for deletion errors thathave never been known before. This is an interesting example that can beproved by good use of the combinatorial equation above. Here, we fix aninteger ≤ t < N . Theorem 5.2.
Let g, n be integers and u be a rational number with g ≥ t +1 , n ≥ t + 1 , u := Ngn ≥ . Suppose that sets A, B ⊂ { , , . . . , N } and a map f : A ∪ B → C are set as A := { gl | ≤ l ≤ n, l : even } ,B := { gl | ≤ l ≤ n, l : odd } ,f ( gl ) := s (cid:0) nl (cid:1) n − (cid:0) gnugl (cid:1) . Then, the code Q fA,B is an [ N, t -deletion error-correcting code.Proof. It is clear that A = ∅ , B = ∅ , A ∩ B = ∅ . Hence, it is enough toprove the three conditions (D1), (D2), and (D3) hold by Theorem 3.4.A simple calculation shows that X w ∈ A | f ( w ) | (cid:18) Nw (cid:19) = 12 n − X ≤ l ≤ nl even (cid:18) nl (cid:19) = 1 . P w ∈ B | f ( w ) | (cid:0) Nw (cid:1) = 1 . Therefore (D1) holds.For an integer ≤ k ≤ t , we obtain X w ∈ A | f ( w ) | (cid:18) N − tw − k (cid:19) − X w ∈ B | f ( w ) | (cid:18) N − tw − k (cid:19) = n X l =0 (cid:0) nl (cid:1) n − (cid:0) gnugl (cid:1) (cid:18) gnu − tgl − k (cid:19) ( − l = 0 by the assumption n ≥ t + 1 and Lemma 5.1. Note that the ratio of binomialcoefficients (cid:0) gnu − tgl − k (cid:1) / (cid:0) gnugl (cid:1) is a polynomial in l of order t . On the other hand,it is obvious that P w ∈ A | f ( w ) | (cid:0) N − tw − k (cid:1) = 0 . Therefore, (D2) holds.It is clear that (D3) holds by the assumption g ≥ t + 1 .The quantum code constructed by Theorem 5.2 include some that arealready known, but its tolerance to deletion errors was mentioned here for thefirst time. This code is called a ( g, n, u ) -PI code in the terms of Ouyang [14].Theorem 5.2 claims that a ( g, n, u ) -PI code is a t -deletion error-correctingcode if g ≥ t + 1 , n ≥ t + 1 , and u ≥ . The smallest example is preciselyHagiwara’s 4-qubit code that is a (2 , , -PI code [6]. The following Fact 5.3is a remarkable result about PI codes by Ouyang [14]. Fact 5.3.
For an integer t ≥ , ( g, n, u ) -PI codes correct arbitrary t -qubiterrors if g = n = 2 t + 1 and u ≥ . Fact 5.3 and Theorem 5.2 show that a (2 t + 1 , t + 1 , u ) -PI code withan integer t ≥ is t -qubit and t -deletion error-correcting code. This isthe first example of codes that can correct both deletion errors and anothertype of errors. The smallest example is precisely Ruskai’s 9-qubit code thatis a (3 , , -PI code [20]. It was already known that this code can correct -qubit errors, but it was shown here for the first time that it can also correct -deletion errors. Here, we fix t := 1 and introduce -deletion error-correcting codes. The codesconstructed by following Theorem 5.4 are already known as examples [21] ofthe code construction given by Nakayama and Hagiwara [12], but it is alsoone family of our codes. 10 heorem 5.4. Suppose that two non-empty sets
A, B ⊂ { , , . . . , N } with A ∩ B = ∅ satisfy followings: • w ∈ A ⇒ N − w ∈ A , • w ∈ B ⇒ N − w ∈ B , • For any integers w , w ∈ A ∪ B , w = w = ⇒ | w − w | > . and a map f : A ∪ B → C are set as f ( w ) := q P w ′∈ A ( Nw ′ ) w ∈ A, q P w ′∈ B ( Nw ′ ) w ∈ B. Then, the code Q fA,B is an [ N,
1] 1 -deletion error-correcting code.Proof.
It is clear that (D1) and (D3) hold by the assumptions. Hence weshow that (D2) holds. By the assumption, we have X w ∈ A (cid:18) N − w − (cid:19) = X w ∈ A (cid:18) N − w − (cid:19) , X w ∈ A (cid:18) N − w − (cid:19) + X w ∈ A (cid:18) N − w − (cid:19) = X w ∈ A (cid:18) Nw (cid:19) . Similarly, the same equations for B are obtained. Hence, X w ∈ A | f ( w ) | (cid:18) N − tw − k (cid:19) − X w ∈ B | f ( w ) | (cid:18) N − tw − k (cid:19) = P w ∈ A (cid:0) N − w − k (cid:1)P w ′ ∈ A (cid:0) Nw ′ (cid:1) − P w ∈ B (cid:0) N − w − k (cid:1)P w ′ ∈ B (cid:0) Nw ′ (cid:1) = 12 −
12= 0 holds for any integer ≤ k ≤ . Therefore, (D2) holds.11 Generalization
In this section, we discuss constructions of [ N, K ] t -deletion error-correctingcodes for any positive integer K . Let L be a positive integer and A , A , . . . ,A L − ⊂ { , , . . . , N } be mutually disjoint non-empty sets and f : S L − i =0 A i → C be a map which satisfy the following three conditions:(D1)* For any integer i ∈ { , , . . . , L − } , X w ∈ A i | f ( w ) | (cid:18) Nw (cid:19) = 1 . (D2)* For any integers ≤ k ≤ t and i, j ∈ { , , . . . , L − } , X w ∈ A i | f ( w ) | (cid:18) N − tw − k (cid:19) = X w ∈ A j | f ( w ) | (cid:18) N − tw − k (cid:19) = 0 . (D3)* For any integers w , w ∈ S L − i =0 A i , w = w = ⇒ | w − w | > t. Let us define an encoder as a linear map
Enc f { A i } : C L → C ⊗ N . For aquantum state | ψ i = P L − i =0 α i | i i ∈ C L , where | i , | i , . . . , | L − i is thestandard orthogonal basis of C L , Enc f { A i } maps the state | ψ i to the followingstate | Ψ i , | Ψ i := L − X i =0 P x ∈{ , } N wt( x ) ∈ A i α i f (wt( x )) | x i ! . Note that this encoder is an extension of Definition 4.1. We claim that theimage of
Enc f { A i } is a t -deletion error-correcting code for an integer ≤ t < N .We can use the same method as in Section 4 for the proof. For the case L = 2 K , we obtain a [ N, K ] t -deletion error-correcting code.Although we do not discuss it in detail here, we can construct [ N, K ] single-deletion error-correcting codes with any integer K ≥ by extendingTheorem 5.4. But no example of [ N, K ] t -deletion error-correcting codes with K ≥ and t ≥ has been found to date.12 Conclusion
This paper gave a construction of permutation-invariant quantum codes fordeletion errors. In particular, the codes given in Theorem 5.2 contain thefirst example of quantum codes that can correct two or more deletion errorsand the first example of codes that can correct both multiple-qubit errorsand multiple-deletion errors.
Acknowledgment
The research has been partly executed in response to support by KAKENHI18H01435.
Appendix A Proof of Lemma 4.2
Proof.
By Equations (2) and (3), it is clear that | Ψ i = X y ∈{ , } t (cid:0) | y i ⊗ | Ψ wt( y ) i (cid:1) for any integer ≤ t < N . By the permutation-invariance of | Ψ i and thedefinition of the partial trace, D P ( | Ψ ih Ψ | ) = Tr ◦ · · · ◦ Tr | {z } t times ( | Ψ ih Ψ | )= X y ∈{ , } t | Ψ wt( y ) ih Ψ wt( y ) | = t X k =0 (cid:18) tk (cid:19) | Ψ k ih Ψ k | holds for any deletion position P ⊂ [ N ] . Appendix B Proof of Lemma 4.3
Proof.
For an integer ≤ k ≤ t , suppose that | U k i := X w ∈ A P x ∈{ , } N − t wt( x )= w − k f ( w ) | x i ! , | U k i := X w ∈ B P y ∈{ , } N − t wt( y )= w − k f ( w ) | y i ! .
13y the condition (D2), h U k | U k i = h U k | U k i 6 = 0 . Set l k ∈ R and | u k i , | u k i ∈ C ⊗ ( N − t ) as l k := q h U k | U k i , | u k i := | U k i l k , | u k i := | U k i l k . Then, h u k | u k i = h u k | u k i = 1 holds. Hence, we have | Ψ k i = α | U k i + β | U k i = l k ( α | u k i + β | u k i ) by Equations (3) and (4), In the case ( k , b ) = ( k , b ) , we obtain h u k b | u k b i =0 by the condition (D3). Appendix C Proof of Lemma 4.5
Proof.
In the case k ∈ { , , . . . , t } , we have p ( k ) = Tr( P k D P ( | Ψ ih Ψ | ))= Tr X x ∈ W k | x ih x | t X k ′ =0 (cid:18) tk ′ (cid:19) | Ψ k ′ ih Ψ k ′ | ! = Tr (cid:18)(cid:18) tk (cid:19) | Ψ k ih Ψ k | (cid:19) = Tr (cid:18)(cid:18) tk (cid:19) l k ( α | u k i + β | u k i )( α h u k | + β h u k | ) (cid:19) = (cid:18) tk (cid:19) l k (cid:0) | α | h u k | u k i + | β | h u k | u k i (cid:1) = (cid:18) tk (cid:19) l k by Lemmas 4.2 and 4.3. In the case k = ∅ , it is clear that p ( ∅ ) =Tr( P ∅ D P ( | Ψ ih Ψ | )) = 0 .Given that outcome k ∈ { , , . . . , t } occurred, by Lemma 4.2, the quan-tum state immediately after the measurement is P k D P ( | Ψ ih Ψ | ) P k Tr( P k D P ( | Ψ ih Ψ | )) = P k (cid:0)P tk ′ =0 (cid:0) tk ′ (cid:1) | Ψ k ′ ih Ψ k ′ | (cid:1) P k (cid:0) tk (cid:1) l k = (cid:0) tk (cid:1) | Ψ k ih Ψ k | (cid:0) tk (cid:1) l k = 1 l k | Ψ k ih Ψ k | . eferences [1] Jacob Biamonte, Peter Wittek, Nicola Pancotti, Patrick Rebentrost,Nathan Wiebe, and Seth Lloyd. Quantum machine learning. Nature ,549:195–202, 2017.[2] A Robert Calderbank and Peter W Shor. Good quantum error-correcting codes exist.
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