Controlled jump in the Clifford hierarchy

Controlled jump in the Cliffo rd hiera rchy Yichen Xu ( 许 轶 臣 ) and Xiao W ang ( 王 骁 ) Depa rtment of Physics, Co rnell University , Ithaca, NY, USA W e develop a simple and systematic route to higher levels of the qubit Clifford hierarc h y b y coherently controlling Clifford op erations. Our approac h is based on P auli p eriodicity , defined for a Clifford unitary U as the smallest in teger m ≥ 1 suc h that U 2 m is a Pauli op erator up to phase. W e prov e a sharp con trolled-jump rule sho wing that the con trolled gate C U lies strictly in lev el m + 2 of the hierarc hy , and equiv alently that C U lies in lev el k if U 2 k − 2 is P auli while no smaller p ositiv e p o wer of U is Pauli. W e further quantify the resources required to realize large lev el jumps in the Clifford hierarc hy by proving an essentially tight upper b ound on Pauli p erio dicit y as a function of the n um b er of qubits, whic h implies that accessing high hierarch y levels through con trolled Cliffords requires a num b er of target qubits that gro ws exp onentially with the desired lev el. W e complement this limitation with explicit infinite families of P auli-p erio dic Cliffords whose controlled v ersions achiev e asymptotically optimal jumps. As an application, we propose a proto col for preparing logical catalyst states that enable logical Z 1 / 2 k phase gates via phase kic kbac k from a single jumped Clifford. Contents 1 In tro duction 2 2 P erio dic Clifford gates and con trolled jump 3 2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 The con trolled jump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Prop erties of the Jump ed Cliffords . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4 The qubit resource for controlled jumps . . . . . . . . . . . . . . . . . . . . . . 10 3 Examples of P auli-p erio dic and jump ed Cliffords 12 3.1 Controlled Clifford permutations . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 Optimal Jump ed Cliffords: Con trolled S X -CNOT- H Strings . . . . . . . . . . . 15 4 Application: catalyzed logical phase gate 15 5 Conclusion and Op en Problems 18 A Pro of of Prop osition 9 23 Yichen Xu ( 许 轶 臣 ) : yx639@co rnell.edu 1 1 Intro duction F ault-tolerant quantum computation naturally divides in to t wo regimes: op erations that are protected by stabilizer structure alone, and those that require additional non-stabilizer re- sources. Clifford circuits o ccupy the first regime: they preserve Pauli op erators under con- jugation, admit efficient classical simulation, and arise as the default logical gates for large classes of stabilizer codes. Univ ersalit y is achiev ed b y supplemen ting them with state injection and adaptive Clifford pro cessing. The qubit Clifford hierarch y , introduced by Gottesman and Ch uang [ 1 ], gives a precise stratification of this landscap e as a nested family C 1 ⊂ C 2 ⊂ · · · defined b y the action on Paulis: C 1 is the Pauli group, C 2 is the Clifford group, and higher lev els contain structured non-Clifford op erations implementable by gate telep ortation given suitable resource states [ 1 – 3 ]. This hierarc h y is not merely a definition, but a guide for where ov erhead enters fault- toleran t implemen tations: non-Clifford gates are indisp ensable for universalit y and typi- cally dominate cost [ 2 , 4 – 8 ], y et while standard syn thesis from a fixed univ ersal set suc h as Clifford+ T is possible (appro ximately or exactly) [ 9 – 14 ], man y useful primitiv es naturally sit ab ov e the third level (e.g., m ulti-con trolled phases and fine-grained Z rotations), so di- rect access to higher-lev el structure can yield sa vings in T coun t, depth, and magic-state o v erhead. A t the same time, transversal realizations of logical gates are fundamentally con- strained by the Eastin–Knill theorem [ 15 ]. Moreo v er, lo cality-preserving logical gates in D spatial dimensions are constrained within the D th level of the Clifford hierarch y [ 16 , 17 ]. This motiv ates the construction of error correction co des that p ossess transv ersal gates in higher Clifford hierarch y [ 18 – 21 ]. Mean while, the global structure of C k on many qubits re- mains p o orly understo o d, and most progress has come from isolating sp ecial families, namely semi-Cliffords [ 22 ], generalized semi-Cliffords and structural results for level three [ 23 – 26 ], diagonal hierarch y gates [ 27 , 28 ], and p erm utation gates at level three [ 29 ]. These analyses, while v aluable, do not address a ubiquitous algorithmic mec hanism: the addition of coheren t con trol to an existing gate. Con trolled unitaries app ear in phase estimation, conditional logic, arithmetic subroutines, and measurement-based gadgets. F rom a circuit viewp oint, adding a con trol qubit app ears to b e a mild mo dification, but its effect on hierarch y level can b e dramatic and is far from auto- matic. Two natural questions arise for controlling a unitary U . First, when do es controlled- U b elong to the Clifford hierarch y at all? Second, when it do es, what is the exact lev el con tain- ing it? Anderson and W eipp ert derived strong necessary conditions for controlled gates to lie in the qubit Clifford hierarch y , providing evidence that con trolled gates form a highly con- strained subclass [ 30 ]. In a complemen tary special case, Surti, Daguerre and Kim sho wed that if a Clifford squares to a Pauli string then its controlled version lies in the third level [ 31 ]. In parallel, Ref. [ 32 ] analyzed optimized depth for controlled-Clifford constructions within Clifford+ T compilation, emphasizing that syn thesis costs and hierarc hy level are related but distinct notions. What has b een missing is a broad and sharp criterion that simultaneously answ ers b oth questions for controlled targets and quantifies the resources required to ac hieve large level jumps. In this w ork w e initiate a systematic study of generating higher-level Clifford-hierarch y gates via con trolled-Clifford op erations. W e introduce P auli p erio dicity , whic h is defined as the least n umber of rep eated squarings needed for a Clifford unitary to b ecome a Pauli op erator up to phase. Our main structural result gives an exact controlled jump rule: if a Clifford gate 2 U has Pauli p erio dicity m , then the controlled- U gate lies strictly in the ( m + 2) th level of the hierarc h y , meaning it is contained in C m +2 but not in C m +1 . This turns the existence of a P auli p o w er, whic h previously app eared as a necessary condition, in to a complete classification for controlled Cliffords. It also subsumes the kno wn third-lev el characterization as the case m = 1 [ 31 ]. Ha ving iden tified the exact hierarc h y lev el of controlled- U in terms of the P auli p erio dicit y of U , the natural next question is ho w large a P auli p erio dicit y , and hence how high a hierarch y lev el, can b e attained with a register of n qubits. Using the binary symplectic represen tation of Clifford op erators, we prov e a tight upp er b ound on Pauli p erio dicit y in terms of the n um b er of qubits. Combining this b ound with the controlled jump rule yields an exp onential low er b ound on the num b er of target qubits required to realize controlled gates that first app ear in high hierarc h y lev els. This explains wh y large hierarc h y jumps cannot b e obtained on small registers by simply adding controls. W e complement these limitations with explicit constructions that achiev e the extreme b eha vior. W e prop ose a family of p erio dic Clifford gates that saturate the qubit low er b ound and therefore realize asymptotically optimal controlled jumps. W e also sho w that the resulting jump ed Cliffords admit exact decomp ositions o ver Clifford+ T , whic h connects the hierarch y classification to practical compilation. Finally , w e connect the algebraic results to a fault-toleran t application in higher-order phase resources. Drawing on our algebraic results, w e presen t a proto col for preparing a logical catalyst state that enables logical Z 1 / 2 k phase gates. This provides a concrete route b y which controlled-Clifford structure can b e used to access fine-grained phase gates in a fault-toleran t fashion [ 1 – 3 ]. 2 P erio dic Cliffo rd gates and controlled jump 2.1 Prelimina ries T o set the stage for our discussion, we review some useful definitions and facts ab out the qubit Clifford hierarc h y . Definition 1 ( n -qubit Pauli group) . L et n ≥ 1 . Define the single-qubit Pauli op er ators X = 0 1 1 0 ! , Y = 0 − i i 0 ! , Z = 1 0 0 − 1 ! . (1) The n -qubit Pauli gr oup is the sub gr oup P n ⊂ U (2 n ) gener ate d by { i I , X j , Y j , Z j } n j =1 , wher e X j := I ⊗ ( j − 1) ⊗ X ⊗ I ⊗ ( n − j ) (and similarly for Y j , Z j ). Equivalently, P n =  ω P 1 ⊗ · · · ⊗ P n : ω ∈ {± 1 , ± i } , P j ∈ { I , X , Y , Z }  . (2) Definition 2 (Qubit Clifford hierarc h y [ 1 ]) . Fix n ≥ 1 . L et P n denote the n -qubit Pauli gr oup. The n -qubit Cliffor d hier ar chy is the neste d family of subsets {C ( n ) k } k ≥ 1 define d r e cursively by C ( n ) 1 := P n , C ( n ) k +1 :=  U ∈ U (2 n ) : U P U † ∈ C ( n ) k for al l P ∈ P n  . (3) In p articular, the se c ond level C ( n ) 2 is the Cliffor d gr oup. W e denote the c ol le ction of al l unitaries in the n qubit hier ar chy as C H . 3 Prop osition 1 (Basic properties of C H ) . Fix n ≥ 1 . (1) (Neste dness) F or al l k ≥ 1 , one has C ( n ) k ⊆ C ( n ) k +1 . F or al l m ≥ n , C ( n ) k ⊆ C ( m ) k . (2) (Gr oup pr op erty) The se c ond level C ( n ) 2 is a gr oup under multiplic ation (the Cliffor d gr oup). F or k ≥ 3 , C ( n ) k is in gener al not a gr oup. (3) (Cliffor d invarianc e) If U ∈ C ( n ) k and C 1 , C 2 ∈ C ( n ) 2 , then C 1 U C 2 ∈ C ( n ) k . (4) (Gener ating set for C ( n ) 2 ) The Cliffor d gr oup C ( n ) 2 is gener ate d by the S gate S := 1 0 0 i ! (4) the Hadamar d gate H := 1 √ 2 1 1 1 − 1 ! (5) and the two-qubit c ontr ol le d-NOT gate CNOT := | 0 ⟩⟨ 0 | ⊗ I + | 1 ⟩⟨ 1 | ⊗ X . (6) (5) (Cliffor d+ T cir cuit) L et T := diag(1 , e i π / 4 ) (7) denote the single-qubit T gate. F or unitaries in levels k ≥ 3 , Cliffor d+ T is appr oximately universal in the sense that for any V ∈ U (2 n ) and any ε > 0 , ther e exists a Cliffor d+ T cir cuit that ε -appr oximates V with depth p olylog (1 /ε ) [ 9 , 10 ]. (6) (Single-qubit phase gates in the hier ar chy) F or k ≥ 0 , the single-qubit phase gate Z 1 / 2 k := diag  1 , e i π / 2 k  (8) lies in the ( k + 1) st level C (1) k +1 (with the c onvention C (1) 1 = P 1 ) [ 14 , 27 ]. Definition 3 (Con trolled unitary) . L et n ≥ 1 and let U ∈ U (2 n ) b e an n -qubit unitary. The c ontr ol le d- U gate is the ( n + 1) -qubit unitary C U := | 0 ⟩⟨ 0 | ⊗ I 2 n + | 1 ⟩⟨ 1 | ⊗ U ∈ U (2 n +1 ) , (9) wher e the first qubit is the c ontr ol. Equivalently, in the c omputational b asis or der e d as {| 0 ⟩ ⊗ | x ⟩ , | 1 ⟩ ⊗ | x ⟩} x ∈{ 0 , 1 } n , C U has the blo ck-diagonal form C U = I 2 n 0 0 U ! . (10) Prop osition 2. L et P ∈ P n b e an n -qubit Pauli, then C P ∈ C ( n +1) 2 . 4 2.2 The controlled jump Our main study concerns a basic but subtle question: given an n -qubit unitary U , when do es adding a control promote it to a higher level of the Clifford hierarc h y? Concretely , we seek ho w the hierarc h y level of C U relates to that of U . A general necessary condition for C U to b e in the Clifford hierarch y was pro v ed by An- derson and W eipp ert [ 30 ]. Theorem 1 ([ 30 ] Corollary 2.4.1) . A c ontr ol le d unitary C U is in C H only if U ∈ C H and U 2 m = P for some m ∈ N and P ∈ P n is some n -qubit Pauli. The pro of relies on a few elemen tary iden tities for controlled unitaries, whic h w e record here for later use. Lemma 1 (Prop erties of con trolled gates) . L et U ∈ U (2 n ) b e a unitary on the n -qubit tar get r e gister. (1) (Distribution law of c ontr ol le d gates) L et V ∈ U (2 n ) . Then C ( U V ) = C U C V . (11) Equivalently, in cir cuit form (with gates applie d fr om left to right), n U V = n V U . (12) (2) (Contr ol- X c onjugation) L et X denote the Pauli- X on the c ontr ol qubit. Then C U ( X ⊗ I 2 n ) = ( I 2 ⊗ U † ) C ( U 2 ) ( X ⊗ I 2 n ) C U. (13) This identity c an b e visualize d as the cir cuit e quivalenc e n X U = n X U U 2 U † . (14) (3) (Pushing a tar get unitary thr ough a c ontr ol) F or any V ∈ U (2 n ) acting on the tar get r e gister, C U ( I 2 ⊗ V ) = C ( U V U † V † ) ( I 2 ⊗ V ) C U. (15) Equivalently, in the cir cuit diagr am, we have n V U = n U V U V U † V † . (16) The relations can b e v erified straightforw ardly . 5 Lemma 2 (Controlled-block-diagonal unitary) . L et A, B ∈ U (2 n ) . F or every r ≥ 1 , if a c ontr ol le d-blo ck-diagonal unitary D = | 0 ⟩⟨ 0 | ⊗ A + | 1 ⟩⟨ 1 | ⊗ B (17) lies in C ( n +1) r , then b oth A and B lie in C ( n ) r . Pr o of. F or an y P ∈ P n one has D ( I 2 ⊗ P ) D † = | 0 ⟩⟨ 0 | ⊗ ( AP A † ) + | 1 ⟩⟨ 1 | ⊗ ( B P B † ) , (18) whic h lies in C ( n +1) r − 1 b y definition. Applying the same reasoning recursively for r − 1 , r − 2 , . . . , 1 sho ws that for each P the op erators AP A † and B P B † ev en tually land in P n after r − 1 steps, i.e., A, B ∈ C H . More precisely , A, B ∈ C ( n ) r . Pr o of of The or em 1 . W e argue b y iterating the commutator-t yp e identit y ( 13 ). Fix k ≥ 1 and supp ose that C U ∈ C ( n +1) k . Let X denote the Pauli- X on the control qubit and set W 1 := ( X ⊗ I 2 n ) C U ( X ⊗ I 2 n ) C U † . (19) By Definition 2 , conjugation by an elemen t of C ( n +1) k maps Paulis to C ( n +1) k − 1 . Hence W 1 ∈ C ( n +1) k − 1 . More generally , define recursiv ely for m ≥ 1 W m +1 := ( X ⊗ I 2 n ) W m ( X ⊗ I 2 n ) W † m . (20) Using again the definin g prop erty of the hierarch y together with the Clifford-inv ariance prop- ert y (Prop osition 1 (3)), w e obtain the level b ound W m ∈ C ( n +1) k − m for all 1 ≤ m ≤ k − 1 . (21) On the other hand, Eq. ( 13 ) implies b y a straigh tforw ard induction that W m is diagonal in the con trol qubit and equals W m = U − 2 m − 1 0 0 U 2 m − 1 ! . (22) In particular, taking m = k − 1 gives W k − 1 ∈ C ( n +1) 1 = P n +1 . Since W k − 1 has the abov e blo c k-diagonal form, this forces U 2 k − 2 to b e an n -qubit Pauli (up to an o v erall phase), i.e., U 2 k − 2 = ± U 2 k − 2 = P for some P ∈ P n . It remains to justify the condition U ∈ C H . Fix an y target P auli P ∈ P n . Since I 2 ⊗ P ∈ P n +1 and C U ∈ C ( n +1) k , we hav e C U ( I 2 ⊗ P ) C U † ∈ C ( n +1) k − 1 . (23) But C U is blo c k diagonal in the con trol qubit, so the conjugate is explicitly C U ( I 2 ⊗ P ) C U † = | 0 ⟩⟨ 0 | ⊗ P + | 1 ⟩⟨ 1 | ⊗ ( U P U † ) . (24) Applying Lemma 2 to the blo ck-diagonal op erator in Eq. ( 23 ) yields U P U † ∈ C ( n ) k − 1 for all P aulis P ∈ P n . Therefore U ∈ C ( n ) k ⊆ C H . 6 While Theorem 1 constrains when C U can lie in C H , it do es not by itself iden tify the hierarc h y lev el of C U . Indeed, the ab ov e argument only shows that some p ow er U 2 m m ust ev en tually fall in to the Pauli group; because the hierarch y is nested, this can happ en strictly b efore the iteration reaches m = k − 1 , and therefore the b ound obtained from the pro of need not b e tight. A complementary characterization in the Clifford case U ∈ C ( n ) 2 w as obtained by Surti, Daguerre, and Kim [ 31 ]: Theorem 2 ([ 31 ] Lemma 1) . L et U ∈ C ( n ) 2 b e a Cliffor d unitary. Then C U ∈ C ( n +1) 3 if and only if U 2 = P for some Pauli P ∈ P n . Pr o of. ( C U ∈ C ( n +1) 3 = ⇒ U 2 = P ) This follows from Theorem 1 . ( U 2 = P = ⇒ C U ∈ C ( n +1) 3 ) Assume U ∈ C ( n ) 2 and U 2 ∈ P n . T o sho w C U ∈ C ( n +1) 3 , it suffices by Definition 2 to v erify that for every Pauli op erator Q ∈ P n +1 , the conjugate C U Q C U † is a Clifford, i.e., lies in C ( n +1) 2 . Ev ery ( n + 1)-qubit Pauli can b e written (up to phase) as a pro duct of a Pauli on the con trol and a P auli on the target, so it is enough to chec k generators X ⊗ I 2 n , Z ⊗ I 2 n , and I 2 ⊗ P with P ∈ P n . First, Z ⊗ I 2 n comm utes with C U . Next, applying Eq. ( 13 ) and using U − 2 = ( U 2 ) † ∈ P n , we obtain C U ( X ⊗ I 2 n ) C U † = ( X ⊗ I 2 n ) ( I 2 ⊗ U − 2 ) C ( U 2 ) . (25) Since U 2 is Pauli, the controlled-P auli C ( U 2 ) is a Clifford (it is a m ulti-controlled Pauli phase/bit-flip), and I 2 ⊗ U − 2 is also P auli; therefore the right-hand side is a pro duct of Cliffords and hence lies in C ( n +1) 2 . Finally , for any P ∈ P n , Eq. ( 15 ) gives C U ( I 2 ⊗ P ) C U † = C ( U P U † P † ) ( I 2 ⊗ P ) . (26) Because U is Clifford, U P U † ∈ P n , hence the commutator U P U † P † is again P auli, so C ( U P U † P † ) ∈ C ( n +1) 2 from Prop osition 2 . Multiplying by I 2 ⊗ P ∈ C ( n +1) 1 preserv es member- ship in C ( n +1) 2 . This prov es that C U conjugates P n +1 in to C ( n +1) 2 , and therefore C U ∈ C ( n +1) 3 . Our goal is to extend this result to controlled jumps by an arbitrary n umber of hierarc hy lev els and to determine the exact hierarc h y lev el of the controlled gate. T o do so, it is con v enien t to record the precise momen t at whic h a unitary b ecomes Pauli under rep eated squaring. Definition 4 (P auli p erio dicity) . L et U ∈ U (2 n ) . W e say that U is m -Pauli-p erio dic if U 2 m ∈ P n and U 2 t / ∈ P n for al l t < m, (27) wher e memb ership in P n is understo o d up to an over al l phase {± 1 , ± i } . 7 Equiv alently , we can define the P auli p erio dicity as the least n umber of times that a unitary U has to be squared in order to become a Pauli op erator: m = min  t ≥ 0 : U 2 t ∈ P n  . (28) Note that, even though every Pauli matrix squares to the iden tit y , the actual order of an m -P auli-p erio dic unitary U can b e ord( U ) = 2 m + q , where q = 0 , 1 , 2 is the log 2 -p erio dicity of the Pauli operator U 2 m . This depends on whether the Pauli operator U 2 m is identit y , Pauli string with ± 1 phase, or Pauli string with ± i phase, resp ectiv ely . With this notion, we can state the exact hierarc h y lev el of the con trolled unitary . Theorem 3 (Controlled jump criterion) . L et U ∈ C ( n ) 2 b e a Cliffor d unitary. Then the c ontr ol le d gate C U lies strictly in level m + 2 if U has Pauli p erio dicity m . That is, C U ∈ C ( n +1) m +2 \ C ( n +1) m +1 . Theorem 3 is a refinement of the necessary condition in Theorem 1 and a generalization of Theorem 2 : here w e not only require that some pow er of U lies in the P auli group, but sho w that the minimal such p ow er precisely determines the hierarch y level of C U . As we will see in the pro of b elo w, the assumption U ∈ C ( n ) 2 is essential here, without which we cannot pinp oint the level of C U . Pr o of. W e pro v e the mem b ership and strictness statements separately . Step 1: C U ∈ C ( n +1) m +2 . By Definition 2 , it suffices to show that for every ( n + 1)-qubit P auli Q ∈ P n +1 , the conjugate C U Q C U † lies in C ( n +1) m +1 . W e prov e this via induction. First of all, the statement is clearly true for m = 1 from the Theorem 2 . No w assuming the statement is true up to m − 1. That is, for ev ery V that has Pauli perio dicit y m ′ < m , C V lies strictly in lev el m ′ + 2. W e no w v erify the claim for C U where U has P auli p erio dicity m . T o this end, we note that every ( n + 1)-qubit P auli is (up to phase) one of Z ⊗ I 2 n , I 2 ⊗ P , X ⊗ P , ( X Z ) ⊗ P , (29) with P ∈ P n . W e handle these cases one by one. • Z ⊗ I 2 n comm utes with C U . • F or I 2 ⊗ P , since U is Clifford w e ha ve U P U † ∈ P n , and Eq. ( 26 ) sho ws C U ( I 2 ⊗ P ) C U † is a pro duct of a con trolled-Pauli and a Pauli; hence it is a Clifford, i.e., it lies in C ( n +1) 2 ⊆ C ( n +1) m +1 . • F or X ⊗ I 2 n , Eq. ( 13 ) implies C U ( X ⊗ I 2 n ) C U † = ( X ⊗ I 2 n ) ( I 2 ⊗ U − 2 ) C ( U 2 ) . (30) Because U is Clifford, U − 2 is also Clifford, so ( X ⊗ I 2 n )( I 2 ⊗ U − 2 ) ∈ C ( n +1) 2 from group prop ert y of Cliffords. By Proposition 1 (3), the hierarc hy level of the righ t- hand side is the same as that of C ( U 2 ). Set V := U 2 . Since U has P auli p erio dicity m , V has Pauli perio dicity m − 1. By the induction hypothesis applied to V , we hav e C V ∈ C ( n +1) ( m − 1)+2 = C ( n +1) m +1 . (31) 8 Hence C ( U 2 ) = C V ∈ C ( n +1) m +1 , and therefore Eq. ( 30 ) shows C U ( X ⊗ I ) C U † is in the lev el m + 1. • F or X ⊗ P , w e write it as X ⊗ P = ( I 2 ⊗ P )( X ⊗ I 2 n ). Then, using Eq. ( 11 ), w e hav e C U ( X ⊗ P ) C U † =  C U ( I 2 ⊗ P ) C U †  C U ( X ⊗ I 2 n ) C U †  , (32) whic h is a pro duct of a Clifford gate (from Eq. ( 26 )) and an element of C ( n +1) m +1 . Hence, from Prop osition 1 (3), it lies in C ( n +1) m +1 as well. • The case ( X Z ) ⊗ P follo ws similarly since C U commutes with Z ⊗ I . All of these results pro ve that C U ∈ C ( n +1) m +2 . Step 2: C U / ∈ C ( n +1) m +1 . If C U ∈ C ( n +1) m +1 , then the iteration in the pro of of Theorem 1 (with k = m + 1) w ould imply that U 2 m − 1 is Pauli (up to phase), contradicting the assumption that U has P auli perio dicity m . Com bining the t w o steps yields C U ∈ C ( n +1) m +2 \ C ( n +1) m +1 . The pro of ab ov e crucially uses Clifford inv ariance (Prop osition 1 (3)) together with the fact that Cliffords conjugate Paulis to P aulis. F or general U ∈ C H (not necessarily Clifford), these conditions will fail, and the same argumen t do es not determine the exact lev el of C U . W e will briefly discuss controlled jumps from non-Clifford inputs in Sec. 5 . 2.3 Prop erties of the Jump ed Cliffords Motiv ated by Theorem 3 , w e isolate the class of con trolled gates obtained from controlling P auli-p erio dic Cliffords. Definition 5 (Jump ed Cliffords) . A jump e d Cliffor d is a c ontr ol le d unitary C U wher e the tar get unitary U ∈ C ( n ) 2 is Pauli-p erio dic. Prop osition 3 (Inv erse of jumped Cliffords) . The inverse of every jump e d Cliffor d is again a jump e d Cliffor d in the same level of the Cliffor d hier ar chy. This follows immediately from closure of the Clifford group under inv ersion, the distribu- tion la w for con trolled gates, and the fact that the inv erse of a unitary has the same perio dicity as the original. This closure under in version is a useful feature of jump ed Cliffords. F or k ≥ 3 , the higher lev els C ( n ) k are not groups in general, so taking an inv erse need not preserv e the hierarc h y level for an arbitrary k th-level unitary . Prop osition 4 (Controlled jump of tensor pro duct Cliffords) . L et U = U 1 ⊗ U 2 wher e U i ∈ C ( n i ) 2 ar e Pauli-p erio dic Cliffor ds with Pauli p erio dicities m i (Definition 4 ). Then U is Pauli- p erio dic with p erio dicity m = max { m 1 , m 2 } , (33) and c onse quently the c ontr ol le d gate C ( U 1 ⊗ U 2 ) lies in C ( U 1 ⊗ U 2 ) ∈ C ( n 1 + n 2 +1) m +2 . (34) 9 In fact, since ( U 1 ⊗ U 2 ) 2 t = U 2 t 1 ⊗ U 2 t 2 , the smallest t for which this b ecomes a Pauli (up to phase) is exactly t = max { m 1 , m 2 } , b ecause U 2 t i is Pauli iff t ≥ m i . With m = max { m 1 , m 2 } , Theorem 3 applied to the Clifford U 1 ⊗ U 2 implies that C ( U 1 ⊗ U 2 ) ∈ C m +2 . Prop osition 4 is meaningful in logical computation, where a transv ersal gate ¯ U is implemen ted b y a tensor pro duct of physical Cliffords acting on eac h individual party . It shows that the controlled logical gate C ( ¯ U ) lies in the same lev el of the Clifford hierarch y as eac h physical controlled gate C ( U p ) . Prop osition 5 (Jump ed Cliffords are Clifford+ T ) . Every jump e d Cliffor d admits an exact Cliffor d+ T de c omp osition. Pr o of. F rom Prop osition 1 (4), every Clifford U can b e expressed as a pro duct of S gates, Hadamards, and CNOT s. Using the distribution law in Eq. ( 11 ), the corresp onding con- trolled gate C U can therefore b e written as a product of gates of the form C ( S ), C ( H ), and C (CNOT). Since eac h factor in such a pro duct has an exact Clifford+ T decomp osition [ 12 ], concatenating these decomp ositions yields an exact Clifford+ T circuit for C U . Prop osition 5 sho ws that every jump ed Clifford can b e implemented exactly using only Clifford+ T op erations. Thus, unlike finer phase gates, ev en when a jump ed Clifford lies in a high level of the Clifford hierarc h y , it do es not require appro ximate synthesis from a universal gate set. This observ ation will b e crucial in our logical phase gate protocol in Sec. 4 . 2.4 The qubit resource fo r controlled jumps Theorem 3 sho ws that achieving a jump to a high level requires a Clifford gate with large P auli p erio dicity . Here we prov e a general upper b ound on how large the Pauli p erio dicity of an n -qubit Clifford can b e, and we give an explicit family of Cliffords that attains this b ound. Theorem 4 (Upp er b ound on Pauli p erio dicity) . L et U ∈ C ( n ) 2 b e a Cliffor d unitary on n qubits. Supp ose that U is m -Pauli-p erio dic (Definition 4 ), i.e., U 2 m ∈ P n up to phase. T hen m ≤ ⌈ log 2 (2 n ) ⌉ . (35) Mor e over, the upp er b ound is tight: for e ach n > 1 ther e exists a Cliffor d unitary U ∈ C ( n ) 2 whose Pauli p erio dicity is exactly ⌈ log 2 (2 n ) ⌉ . Before pro ving Theorem 4 , we recall the standard binary symplectic description of n - qubit Clifford unitaries: a Clifford is determined (up to global phase) b y its action on Pauli op erators, which can b e enco ded by a symplectic matrix ov er F 2 together with a linear phase function. Lemma 3 (Binary matrix representation of Cliffords [ 33 , 34 ]) . L et U ∈ C ( n ) 2 b e an n -qubit Cliffor d unitary. Then ther e exist F ∈ Sp(2 n, F 2 ) and γ ∈ ( Z 4 ) 2 n (36) such that for every Pauli op er ator P ∈ P n written (up to phase) as P ≡ X x Z z , ( x , z ) ∈ ( F 2 ) 2 n , (37) 10 one has U P U † ≡ i ⟨ γ , ( x , z ) ⟩ X x ′ Z z ′ , ( x ′ , z ′ ) = F ( x , z ) , (38) wher e ⟨· , ·⟩ denotes the natur al p airing ( Z 4 ) 2 n × ( F 2 ) 2 n → Z 4 , and ≡ denotes e quality up to an over al l phase. Conversely, any p air ( F , γ ) of this form sp e cifies a Cliffor d unitary up to glob al phase. Lemma 4 (P erio dicity b ound for in v ertible binary matrices) . L et F ∈ SL(2 n, F 2 ) and supp ose that F has 2 -p ower or der, i.e., F 2 t = I (39) for some t ≥ 0 . Then F is unip otent and the nilp otent matrix N := F − I satisfies ( F − I ) 2 n = N 2 n = 0 . (40) Mor e over, F 2 ⌈ log 2 (2 n ) ⌉ = I . (41) Pr o of. Let F 2 b e an algebraic closure of F 2 and let λ b e an eigenv alue of F o v er F 2 . F rom F 2 t = I we obtain λ 2 t = 1. Since λ lies in some finite extension F 2 s , it belongs to the cyclic group F × 2 s of order 2 s − 1 (o dd). Hence the only element of 2-p ow er order in F × 2 s is 1, so λ = 1. Th us all eigen v alues of F are 1, i.e., F is unipotent. Since F is unip otent, its minimal p olynomial has the form ( x − 1) r with 1 ≤ r ≤ 2 n . Equiv alently , writing F = I + N we ha v e N r = 0, hence in particular N 2 n = 0. Ov er characteristic 2, the F rob enius/binomial iden tity giv es ( I + N ) 2 k = I + N 2 k for all k ≥ 0 . (42) Cho osing k = ⌈ log 2 (2 n ) ⌉ yields 2 k ≥ 2 n ≥ r , so N 2 k = 0 and therefore F 2 k = ( I + N ) 2 k = I . Pr o of of The or em 4 . Let F ∈ Sp(2 n, F 2 ) be the binary symplectic matrix of U from Lemma 3 . If U 2 m ∈ P n (up to phase), then conjugation by U 2 m acts trivially on Pauli lab els, hence F 2 m = I . (43) In particular, F has 2-pow er order, so Lemma 4 applies and yields F 2 ⌈ log 2 (2 n ) ⌉ = I . (44) Therefore m ≤ ⌈ log 2 (2 n ) ⌉ . T o sho w the upp er bound is achiev able, w e use the existence of a regular unip otent element of large 2-p ow er order in Sp(2 n, F 2 ). The group Sp(2 n, F 2 ) is of Lie t yp e C n in defining c har- acteristic p = 2. By T esterman’s order formula for regular unip otent elemen ts [ 35 , Eq. 0.4], there exists a regular unip otent element x ∈ Sp(2 n, F 2 ) whose order is the smallest pow er of the characteristic p that is strictly larger than the height of the highest ro ot. F or t yp e C n the Co xeter n umber is h = 2 n , and the height of the highest ro ot is ht( α 0 ) = h − 1 = 2 n − 1 [ 36 ]. Hence ord( x ) is the smallest 2 k satisfying 2 k > 2 n − 1, i.e., k = ⌈ log 2 (2 n ) ⌉ . (45) 11 Cho ose any Clifford unitary U x ∈ C ( n ) 2 whose induced symplectic action is F = x (suc h a Clifford exists b y Lemma 3 ). Then F 2 k = I implies U 2 k x ∈ P n up to phase. Moreo v er, since ord( x ) = 2 k is exact, w e ha ve F 2 k − 1  = I , and therefore U 2 k − 1 x / ∈ P n . Thus U x has P auli p erio dicity exactly k = ⌈ log 2 (2 n ) ⌉ . In Sec. 3.2 , w e will pro vide a concrete example of a P auli-p erio dic unitary that saturates the b ound in Theorem 4 . Corollary 1 (Qubit low er b ound for a k -level con trolled jump) . L et U ∈ C ( n ) 2 b e an n -qubit Cliffor d and supp ose that C U lies strictly in level k ≥ 4 , i.e., C U ∈ C ( n +1) k \ C ( n +1) k − 1 . Then the tar get unitary U must act nontrivial ly on at le ast n ≥ 2 k − 4 + 1 qubits. Pr o of. If C U is strictly in lev el k , then by Theorem 3 the Clifford U has Pauli p erio dicity m = k − 2. Applying Theorem 4 to U giv es k − 2 = m ≤ ⌈ log 2 (2 n ) ⌉ . (46) This implies 2 n > 2 k − 3 and hence the smallest possible in teger n is 2 k − 4 + 1 for k ≥ 4. 3 Examples of P auli-p erio dic and jump ed Cliffords In this section, w e pro vide a few examples of Pauli-perio dic Cliffords and their jumped Clif- fords. In particular, w e construct p ermutation gates in higher Clifford hierarch y via con trolled Clifford permutations. In addition, w e give the example of an optimal p erio dic Clifford that saturates the lo w er b ound in Corollary 1 . 3.1 Controlled Cliffo rd p ermutations W e first consider control gates of a simple and well-structured subclass of Clifford unitaries: P auli-p erio dic Cliffords that act as p erm utations of computational basis states. Definition 6 (Perm utation gate) . A n n -qubit unitary U is c al le d a p ermutation gate if ther e exists a p ermutation π of { 0 , 1 } n such that U π | a ⟩ = | π ( a ) ⟩ for al l a ∈ { 0 , 1 } n . (47) Prop osition 6 (P olynomial descriptions of a p ermutation gate) . Every p ermutation gate over n qubits c an b e written as U π = X a ∈{ 0 , 1 } n | π ( a ) ⟩ ⟨ a | , (48) wher e π is a p ermutation of { 0 , 1 } n . W riting π ( a ) = ( π 1 ( a ) , . . . , π n ( a )) and viewing a = ( a 1 , . . . , a n ) as a ve ctor over F 2 , e ach c o or dinate function π i : { 0 , 1 } n → { 0 , 1 } admits a unique r epr esentation as a p olynomial in a 1 , . . . , a n over F 2 . Since w e are considering P auli-p erio dic Cliffords, one interesting class is the P auli-p erio dic Clifford p erm utation. 12 Definition 7 (Clifford p erm utation) . A n n -qubit Cliffor d p ermutation is a p ermutation gate that c an b e gener ate d by a se quenc e of CNOT and X gates. F or Clifford p ermutations, we ha ve these useful results: Lemma 5 (Clifford permutations are affine linear [ 29 ], Prop osition 2.15) . L et U b e an n -qubit Cliffor d p ermutation. Then ther e exist a binary matrix M ∈ GL( n, F 2 ) and a fixe d bitstring ϕ ∈ { 0 , 1 } n such that for al l a ∈ { 0 , 1 } n , U | a ⟩ = | M a + ϕ ⟩ , (49) wher e addition and matrix multiplic ation ar e over F 2 (identifying bitstrings with c olumn ve c- tors in F n 2 ). Lemma 5 shows that Clifford permutations are precisely reversible affine maps on bit- strings. T o analyze their Pauli p erio dicity , it is useful to write down the asso ciated binary symplectic representation. Prop osition 7 (Binary symplectic matrix of a Clifford p ermutation) . L et U b e an n -qubit Cliffor d p ermutation with line ar p art M ∈ GL( n, F 2 ) as in L emma 5 . Then the binary sym- ple ctic matrix F ∈ Sp(2 n, F 2 ) of U (L emma 3 ) is blo ck diagonal and e quals F = M 0 0  M − 1  T ! . (50) Pr o of. W e verify this by directly computing the action of U on P auli generators. Let e i b e the i th standard basis v ector. F or eac h i , the operator X i acts as | a ⟩ 7→ | a + e i ⟩ , so U X i U † | M a + ϕ ⟩ = U X i | a ⟩ = U | a + e i ⟩ = | M a + M e i + ϕ ⟩ . (51) Th us U X i U † = X M e i up to phase. Equiv alently , on X -lab els the induced linear map is x 7→ M x . Z -typ e Paulis. F or eac h i , Z i | a ⟩ = ( − 1) e i · a | a ⟩ , hence U Z i U † | M a + ϕ ⟩ = U Z i | a ⟩ = ( − 1) e i · a | M a + ϕ ⟩ . (52) W riting a = M − 1 ( M a + ϕ − ϕ ) giv es e i · a = (( M − 1 ) T e i ) · ( M a + ϕ ) + (constant), so U Z i U † equals Z ( M − 1 ) T e i up to an ov erall phase. Therefore on Z -lab els the induced linear map is z 7→ ( M − 1 ) T z . Since conjugation by U sends X -t yp e P aulis to X -type P aulis and Z -type Paulis to Z -type P aulis, there is no X – Z mixing, so the symplectic matrix is blo ck diagonal with blo cks M and ( M − 1 ) T . Corollary 2 (Pauli-perio dicity of Clifford p erm utations) . F or a Cliffor d p ermutation to b e Pauli-p erio dic, the c orr esp onding matrix M must b e unip otent. F urthermor e, the maximum Pauli p erio dicity that c an b e r e ache d by any Cliffor d p ermutation on n qubits is upp er b ounde d by ⌈ log 2 ( n ) ⌉ . 13 Indeed, if U 2 m ∈ P n up to phase, then conjugation b y U 2 m acts trivially on Pauli lab els, and hence the asso ciated symplectic matrix satisfies F 2 m = I . Using Prop osition 7 , we obtain M 2 m = I and  ( M − 1 ) T  2 m = I , and thus also ( M T ) 2 m = I . Therefore M has 2 -pow er order, whic h (o v er F 2 ) implies that M is unip otent. Applying Lemma 4 for the n × n matrix M yields m ≤ ⌈ log 2 n ⌉ , as claimed. Moreov er, this upp er b ound can b e ac hiev ed, for example, b y taking M to b e a single Jordan blo ck with ones on the diagonal and the sup erdiagonal, i.e., M ij = ( 1 , i = j or i = j − 1 , 0 , otherwise . (53) The Clifford p ermutation corresp onding to this M is precisely the CNOT string  Q n − 1 j =1 CNOT j +1 ,j  where the pro duct is taken from right to left. Another simple choice that saturates the upp er b ound in Corollary 2 , whic h can b e realized in a constant depth circuit, is the bric kw ork CNOT circuit:    ⌊ n − 1 2 ⌋ Y j =1 CNOT 2 j +1 , 2 j      ⌊ n 2 ⌋ Y j =1 CNOT 2 j, 2 j − 1   . (54) W e no w consider the properties of the jump ed Clifford p erm utations. Prop osition 8 (Jump ed Clifford p erm utations are quadratic) . L et U b e an n -qubit Cliffor d p ermutation, and let C U denote the c orr esp onding c ontr ol le d gate on n + 1 qubits. Then C U is again a p ermutation gate. Mor e over, identifying c omputational b asis states with bitstrings a = ( a 0 , a 1 , . . . , a n ) ∈ { 0 , 1 } n +1 (wher e a 0 is the c ontr ol bit), the p ermutation induc e d by dr esse d p ermutation gate of the form P C U Q , wher e P , Q ar e Cliffor d p ermutations over the ( n + 1) qubits, c an b e written as a i 7− → Π i ( a 0 , a 1 , . . . , a n ) , i = 0 , 1 , . . . , n, (55) wher e e ach Π i is a p olynomial over F 2 of de gr e e at most 2 . This is straigh tforw ard to see: in a decomp osition P C U Q , the Clifford p ermutation Q first maps the input bits a to new affine-linear bits b = Π Q ( a ) , so that the con trol bit en tering C U is b 0 . The only op eration that can introduce pro ducts is the single controlled gate C U , which can only create cross terms inv olving b 0 ; comp osing with the subsequent Clifford p ermutation P preserv es the degree, so all output co ordinates remain p olynomials of total degree at most 2 in the original input bits a . Prop osition 8 also connects to the p olynomial description of p ermutation gates. Lemma 2.13 of Ref. [ 29 ] sho ws that, for p erm utation gates, the Clifford-hierarc hy lev el upp er-b ounds the degree of a p olynomial description of the in v erse p e rm utation π − 1 . By contrast, the degree of the forw ard map π itself do es not directly control the hierarch y lev el. Jump ed Clifford p erm utations provide a concrete illustration: even though they admit quadratic co ordinate functions, they can still lie in arbitrarily high levels of the Clifford hierarch y given enough qubits. Moreo v er, this hierarch y upp er b ound on deg( π − 1 ) can b e far from tight. Indeed, the in v erse of a dressed jump ed p ermutation is again of the form Q † C ( U † ) P † , where U † is a Clifford permutation with the same P auli p erio dicity . Th us the in v erse of the jump ed p erm utation liv es in the same lev el in the hierarc h y . Mean while, the in v erse map Π − 1 (with 14 Π defined in Eq. 55 ) again admits a quadratic polynomial description, despite the high lev el nature of the original permutation gate. In short, the con trolled Clifford p ermutations demonstrate that low algebraic degree of π or π − 1 do es not preclude a p ermutation gate from b elonging to high levels of the Clifford hierarc h y . 3.2 Optimal Jump ed Cliffo rds: Controlled S X -CNOT- H Strings W e now construct another imp ortant example of Pauli-perio dic Cliffords whose P auli p erio d- icit y achiev es the upp er b ound in Theorem 4 . Definition 8 ( S X − C N O T − H string) . Define S X := H S H as the π / 2 phase gate ab out the X axis. F or n ≥ 2 , we define the S X -CNOT- H string on n qubits to b e the Cliffor d unitary S C H n := S X,n   n − 1 Y j =1 CNOT j,j +1   H 1 , (56) wher e CNOT j,j +1 denotes a CNOT with c ontr ol qubit j and tar get qubit ( j + 1) , the pr o duct is taken fr om right to left (i.e. S X,n is fol lowe d by C N O T n − 1 ,n ), and S X,n (r esp. H 1 ) me ans applying S X (r esp. H ) on qubit n (r esp. qubit 1 ) and identity on al l other qubits. The circuit diagram for S C H n is: . . . . . . 1 H 2 3 . . . n S X . (57) Prop osition 9 ( S X -CNOT- H string saturates the lo w er bound) . The Cliffor d unitary S C H n is Pauli p erio dic, and satur ates the b ound in The or em 4 . That is, for every n > 2 , S C H n has Pauli p erio dicity ⌈ log 2 (2 n ) ⌉ . W e defer the verification of its p erio dicit y to App endix A . As in tuition for the large p erio dicity , note that the CNOT cascade maps a computational basis state to a tw o-comp onen t “cat” sup erp osition supp orted on tw o Hamming-separated bitstrings, and the final S X injects a nontrivial phase b etw een these comp onen ts. Iterating S C H n rep eatedly propagates and mixes these phases across the register, so that the supp ort of a basis state under ( S C H n ) t quic kly spreads to an exp onentially gro wing set of computational basis strings. One therefore exp ects that reac hing a P auli (and hence returning to the identit y up to phase) requires a n um b er of iterations that is linear in n , consistent with the exact order 2 ⌈ log 2 (2 n ) ⌉ implied b y the nilp otency b ound. Mathematically , the H and S X gates spread the Jordan chain in Eq. ( 53 ) to b oth X and Z op erators. 4 Application: catalyzed logical phase gate Con trolled unitaries are cornerstones of quan tum algorithms; in phase estimation and related routines, coherent con trol conv erts an eigenphase of a unitary into a measurable bit string. In 15 our setting, we exploit the same idea in a fault-toleran t con text. Sp ecifically , w e use jump ed Cliffords to prepare logical magic states that are eigenstates of P auli-p erio dic Clifford unitaries with eigenv alue e π i / 2 k . These eigenstates can then b e used as a catalyst to fault-tolerantly implemen t fine logical phase gates ( ¯ Z ) 1 / 2 k Existing measuremen t-and-p ostselection proto cols typically target eigenstates of Clifford unitaries [ 37 – 42 ], and then con v ert them into useful non-stabilizer resources. Canonical ex- amples are | H ⟩ ∝ 1+ H 2 | + ⟩ and | C Z ⟩ ∝ 1+ C Z 1 , 2 2 | ++ ⟩ , where one implements the pro jector by coheren tly con trolling the unitary with an ancilla, measuring the ancilla in the X basis, and p ost-selecting on the +1 outcome. It is tempting to apply the same approach to P auli-p erio dic Cliffords using their control gates, i.e. the jump ed Cliffords; how ever, doing so do es not leverage the higher-level nature of jump ed Cliffords. The reason is that the corresp onding post-selected states already admit exact Clifford+ T preparations. Prop osition 10. L et U b e an m -Pauli-p erio dic Cliffor d and let | ψ ⟩ b e a Pauli eigenstate. Then the state | ψ ⟩ pr oje cte d onto the +1 eigensp ac e of U , namely P 2 m − 1 r =0 U r | ψ ⟩ , c an b e pr ep ar e d exactly using a Cliffor d+ T cir cuit. One wa y to see this is to note that the pro jector P 2 m − 1 r =0 U r can b e generated b y a circuit con taining multiple con trolled- U gates follo wed b y Pauli measuremen ts and p ost-selection; b y Prop osition 5 , con trolled Pauli-perio dic Cliffords admit exact Clifford+ T implementations. T ogether with the P auli basis initialization of | ψ ⟩ , in the ZX-calculus, the corresp onding ZX-diagram for the pro jected state only has spiders whose phases are in teger m ultiples of π / 4 [ 43 , 44 ]. T o implemen t finer phases, we should use jump ed Cliffords in a wa y that exploits their place in the high lev el of the Clifford hierarch y . The key idea is to instead prepare an eigenstate of the Pauli-perio dic Clifford U with eigenv alue e π i / 2 k . Suc h an eigenv alue of U could exist for an m -Pauli-perio dic Clifford with P auli p erio dicity of at least m = k − 1 , in which case the Pauli op erator obtained via squaring the unitary U 2 m = P contains a phase i 1 . This is kno wn as the catalyst state in the literature [ 45 – 47 ]. W e denote this eigenstate by | ψ k ⟩ , which satisfies U | ψ k ⟩ = e π i / 2 k | ψ k ⟩ . (58) Once | ψ k ⟩ is a v ailable, a single application of the jump ed Clifford C U “kicks back” this eigenphase onto the control qubit, thereb y pro ducing the standard single-qubit phase gate Z 1 / 2 k : C U | ϕ ⟩ ⊗ | ψ k ⟩ = ϕ 0 | 0 ⟩ ⊗ | ψ k ⟩ + ϕ 1 | 1 ⟩ ⊗ U | ψ k ⟩ =  ϕ 0 | 0 ⟩ + e π i 2 k ϕ 1 | 1 ⟩  ⊗ | ψ k ⟩ = Z 1 / 2 k | ϕ ⟩ ⊗ | ψ k ⟩ . (59) Here | ϕ ⟩ = ϕ 0 | 0 ⟩ + ϕ 1 | 1 ⟩ is an arbitrary single qubit state. In a fault-tolerant implemen tation, this reduces the problem of fault-tolerantly implemen t Z 1 / 2 k to preparing the logical catalyst eigenstate    ψ k E and applying C U once at the logical 1 W e note that in this case the lev el of the controlled gate C U matches the level of the phase gate Z 1 / 2 k : b oth lie in the ( k + 1)-th lev el from Prop osition 1 (6) and Theorem 3 . Therefore, the effect of the controlled jump has been fully exploited. If P = U 2 m do not con tain phase i , but at least is a non-trivial Pauli string, w e will “w aste” the jumped level by 1. 16 lev el. Imp ortantly , Proposition 5 ensures that the logical gate C U can be implemen ted exactly using Clifford+ T . Thus, if one works with a quan tum error-correcting co de that admits transv ersal implementations of both T and the c hosen p erio dic Clifford U , the en tire routine can b e executed transv ersally without any additional non-stabilizer resource states b eyond those used for transversal T . F or instance, one ma y take U to b e a CNOT string, like the constan t depth one in Eq. ( 54 ) , and use the 3D color co de [ 48 – 50 ], whic h supp orts transversal implemen tations of b oth T and CNOT, since it is a Calderbank-Shor-Steane (CSS) code [ 51 , 52 ]. W e no w describ e a logical-level circuit for preparing    ψ k E . Suppose U is supported on n ph ysical qubits, and that the chosen co de admits a transv ersal implementation of U across n co de blocks. Then the resulting logical op eration factorizes as ¯ U = ⊗ l p =1 U p , (60) where l is the num b er of disjoin t transversal parties, each contains n p = O (1) qubits, and U p denotes the restriction of U to that party . Consequently , a controlled logical op eration b et w een a single ancilla and the enco ded data can b e implemented as a pro duct of smaller con trolled gates, C ( ¯ U ) = l Y p =1 C ( U p ) , (61) as illustrated b y the circuit representation C ( ¯ U ) = P p n p ¯ U = n 1 n 2 n l . . . . . . U 1 U 2 U l . (62) With this decomp osition in hand, w e prepare    ψ k E via quan tum phase estimation (QPE) on ¯ U using ph ysical ancillas. Concretely , we use k + 1 ancilla qubits to resolv e phases in m ultiples of 2 π / 2 k +1 (since the target eigenphase is π / 2 k ), apply controlled p ow ers of ¯ U , and then apply the inv erse quan tum F ourier transform (QFT) and measure the ancillas: k +1 k +1 n P p n p | + ⟩ ⊗ ( k +1) | + ⟩ ⊗ ( k +1) . . . | ψ k ⟩ · · ·    ψ k E QF T QF T E N C Q ( ¯ U ) 2 q − 1 S E Q ( ¯ U ) 2 q − 1 S E . (63) Here w e first prepare the physical eigenstate | ψ k ⟩ of U on n qubits. This can b e done using ph ysical qubit-lev el QPE. W e then enco de it (ENC) into n logical qubits, yielding an initial, 17 noisy realization of    ψ k E . W e then run QPE b etw een the k + 1 ancillas and the enco ded register by applying C (( ¯ U ) 2 q − 1 ) controlled by the k th ancilla (for q = 1 , . . . , k + 1 ), and p ost-select the measurement outcome (1) 2 (i.e. all the digits are 0 except for the first one), whic h corresp onds to the desired eigen v alue e i π 2 k of U . Betw een these controlled-pow er steps, w e interlea ve syndrome extraction (SE) rounds, so that rep eating the routine improv es the fidelit y of the resulting logical eigenstate. Adding the steps of growing the code distance, the en tire proto col can b e p otentially developed into a cultiv ation scheme of the catalyst state | ψ m ⟩ . In eac h round of the logical measuremen t circuit in Eq. ( 62 ) , the depth scales linearly with the num b er of transv eral parties l , since C ( ¯ U ) requires sequen tial action of ph ysical C U gates on eac h transversal party , and each physical U p can b e p erformed in constant depth, if one uses constan t depth realization of perio dic Cliffords, such as the brickw ork CNOT circuit in Eq. ( 54 ) . The depth of ph ysical C U gates can b e further reduced to a constant if one uses a fan-out circuit on b etw een the ancilla and a set of l additional ancillas before the C U gates, similar to the idea in Ref. [ 32 ]. The depth of the (non-fault-tolerant) QFT part of the circuit scales linearly with k . W e point out that the controlled logical gates in the logical QPE, C (( ¯ U ) 2 q − 1 ) , and the phase gates in the physical QFT, all lie in definite levels the Clifford hierarch y for every q , thanks to Prop osition 4 . In addition, our ability to pinp oint the level of the hierarch y for the en tire logical circuit suggests our proto col for catalyst state preparation may b e amenable to more systematic analysis of P auli error propagation which aids to the possibility of classical sim ulation and benchmarking. W e leav e the fault-tolerance analysis and the developmen t of sim ulation technique to future work. 5 Conclusion and Op en Problems This work identifies a simple mec hanism, coherent control of p erio dic Clifford gates, that generates explicit, exactly-synthesizable families of high-lev el Clifford-hierarch y op erations. Our first con tribution is conceptual: we introduced Pauli p erio dicity as an in v ariant tailored to con trolled constructions, capturing the precise point at whic h rep eated squaring of a Clifford collapses to a Pauli. Our main structural result, Theorem 3 , turns this in v ariant in to a sharp con trolled-jump rule: for a Pauli-perio dic Clifford U , the hierarch y lev el of C U is determined exactly b y the P auli p erio dicit y of U . W e also show ed that the resulting jump ed Cliffords enjo y useful closure and compilation prop erties, suc h as stabilit y under in version and under tensor-pro duct targets, and exact Clifford+ T syn thesis. Our second con tribution is quan titativ e: Theorem 4 b ounds P auli perio dicity as a function of the n um b er of qubits, and together with explicit saturating families it yields a tight resource tradeoff. Namely , although con trolled Cliffords can reac h arbitrarily high hierarc h y levels in principle, a jump to level k from Clifford targets requires a num b er of target qubits that gro ws exp onen tially with k . Finally , w e connected the algebraic picture to fault tolerance by prop osing a catalyst-state preparation routine and a catalyzed logical phase-gate proto col based on a single jump ed Clifford. On the level of mathematics, our results raise sev eral natural op en problems. A first direction is to go beyond Clifford inputs and study the c ontr ol le d-jumping p ower of unitaries already in the third and higher lev els of the hierarc hy . In particular: 18 Op en Problem 1. A mong al l the n -qubit unitaries U ∈ C ( n ) k in the k -th level of the Cliffor d hier ar chy with Pauli p erio dicity m (i.e. U 2 m ∈ P n ), what is the highest level of the Cliffor d hier ar chy that C U c an r e ach? This problem is subtle b ecause the outcome can dep end sim ultaneously on three parameters: the perio dicity m , the num b er of target qubits n , and the input lev el k . F or Clifford inputs, Theorem 4 tigh tly couples m to n , but in higher levels this coupling can disapp ear: for example, the single-qubit phase gate Z 1 / 2 k − 1 ∈ C (1) k (Prop osition 1 (6)) has Pauli perio dicit y k − 1 indep endent of n . At the same time, large p erio dicity do es not automatically translate in to a large con trolled jump: Ref. [ 27 ] shows that C Z 1 / 2 k − 1 ∈ C (2) k +1 , i.e., adding a control increases the level by only one. Understanding when coherent control pro duces a genuinely sup er-c onstant lev el increase for non-Clifford inputs w ould rev eal new structure in C k and could p oten tially bypass the exponential qubit o verhead inherent to Clifford targets. Indep enden tly of controlled jumps, the relationship b et w een P auli p erio dicity and the n um b er of qubits is in teresting in its own right. More explicitly: Op en Problem 2. Given a Pauli-p erio dic unitary U ∈ C ( n ) k , what is the lar gest Pauli p eri- o dicity m max that c an b e achieve d by U ? In p articular, for a fixe d high-enough level k , is it p ossible to r e ach m max ∼ poly ( n ) inste ad of log( n ) ? F or the problem to b e meaningful, we must hav e m ≥ k , since we alw a ys hav e a ( k − 1) - P auli-p erio dic unitary in the k -th level, namely Z 1 / 2 k − 1 . This question is highly meaningful giv en the fact that if one drops the P auli-p erio dicity constrain t and instead asks for the maximal order of an n -qubit unitary within a giv en level of the hierarch y , one can obtain the maxim um p erio d of 2 n − 1 (whic h is not a p o wer of 2). A famous example is the linear- feedbac k shift-register (LFSR) construction [ 53 , 54 ], which can b e implemented by Clifford circuits in quan tum computation [ 55 , 56 ]. In contrast, the op en problem poses a m uc h stronger requiremen t that some 2 -p o w er of U collapses to a Pauli. In the con text of con trolled jump, if m max ∼ p oly( n ) for a certain level k , the qubit resource needed for con trolled jump, and subsequen tly the proto col for logical phase gate, could be dramatically reduced. The discussion in Sec. 3.1 raises a related set of questions ab out the complexit y of p er- m utation gates within the Clifford hierarch y . Prop osition 8 shows that the Clifford-dressed jump ed p ermutation gates admit quadratic co ordinate maps and yet can o ccup y arbitrarily high hierarc h y lev els giv en large enough n . Ho w ev er, the quadratic structure realized there is highly restricted: all gen uinely quadratic terms arise only from pro ducts b etw een a single con trol bit and affine-linear functions of the target bits. This motiv ates asking for a principled, algorithmic classification of general quadratic p ermutations. Op en Problem 3. L et U π b e an n -qubit p ermutation gate implementing a bije ction π : { 0 , 1 } n → { 0 , 1 } n . A ssume that b oth π and/or π − 1 admit c o or dinatewise r epr esentations by p olynomials over F 2 of total de gr e e at most 2 (or in gener al a fixe d de gr e e l ). Given such a p olynomial description of π , is ther e an efficient algorithm that c omputes the smal lest k such that U π ∈ C ( n ) k ? On the level of applications, we hop e that Pauli-perio dic Cliffords and their controlled jumps can lead to more efficient preparation and simulation of resource states that can b e used to implemen t high-level logical gates, suc h as the catalyzed phase-gate proto col in Sec. 4 . 19 It is instructive to con trast our setting with approaches suc h as Ref. [ 55 ], whic h realize a fine phase 2 π / (2 k − 1) using eigenstates of p erio dic Cliffords on only k qubits, rather than the n ∼ e k target qubits required to implement a lev el- k con trolled jump using Clifford targets in our proto col. The tradeoff there is that the controlled unitaries in their case lie completely outside the Clifford hierarch y according to Theorem 1 . This can make classical simulation and fault-tolerance analysis substantially more c hallenging for the fault-tolerant version of suc h an algorithm. A ckno wledgement YX ackno wledges supp ort by the NSF through the gran t O A C-2118310. 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A Pro of of Prop osition 9 F rom Lemma 3 , to prov e the P auli p erio dicity of S C H n is ⌈ log 2 (2 n ) ⌉ , we sho w that N n := M n − I 2 n , where M n is the binary symplectic matrix for S C H n , has nilp otency index 2 n . That is, it satisfies N 2 n n = 0 but N r n  = 0 for all 0 ≤ r ≤ 2 n − 1 . The matrix M n has the follo wing blo ck form: M n = A n B n C n D n ! ∈ Sp(2 n, F 2 ) , (64) where the n × n blo c ks A n , B n , C n , D n are given entrywise (for 1 ≤ i, j ≤ n ) by ( A n ) ij = ( 1 , i ≥ j and j > 1 , 0 , otherwise , (65) ( B n ) ij = ( 1 , j = 1 or ( i = j = n ) , 0 , otherwise , (66) ( C n ) ij = ( 1 , i = j = 1 , 0 , otherwise , (67) ( D n ) ij = ( 1 , ( i = j and i > 1) or ( j = i + 1) , 0 , otherwise . (68) W e now compute the action of N n on the symplectic basis. Let { e j } n j =1 b e the standard basis of F n 2 , i.e. ( e j ) k = δ j,k . W e write elements of ( F 2 ) 2 n as column v ectors ( x, z ) with x, z ∈ F n 2 , and define e X j := ( e j , 0) T , e Z j := (0 , e j ) T , (1 ≤ j ≤ n ) . Using the action of M n , the bitstrings are transformed as ( x ′ , z ′ ) T = ( A n x + B n z , C n x + D n z ) T . Therefore, we hav e the following iden tities in ( F 2 ) 2 n . F or the X -basis vectors, N n e X 1 = ( A n − I ) e 1 + ( C n ) e 1 = e X 1 + e Z 1 , (69) N n e X j = (( A n − I ) e j , C n e j ) =  n X i = j +1 e i , 0  = n X i = j +1 e X i , 2 ≤ j ≤ n − 1 , (70) N n e X n = 0 . (71) 23 F or the Z -basis v ectors, N n e Z 1 = ( B n e 1 , ( D n − I ) e 1 ) =  n X i =1 e i , e 1  = e Z 1 + n X i =1 e X i , (72) N n e Z j = ( B n e j , ( D n − I ) e j ) = (0 , e j − 1 ) = e Z j − 1 , 2 ≤ j ≤ n − 1 , (73) N n e Z n = ( B n e n , ( D n − I ) e n ) = ( e n , e n − 1 ) = e X n + e Z n − 1 . (74) W e now show that, starting from e Z n , the rep eated action of N n generates a length- 2 n Jordan chain. F rom ( 74 ) and ( 71 ) we ha ve N k n e Z n = N k − 1 n ( e Z n − 1 + e X n ) = N k − 1 n e Z n − 1 ( k ≥ 2) . Iterating ( 73 ) gives N n − 1 n e Z n = e Z 1 . (75) Applying ( 72 ) then yields N n n e Z n = N n e Z 1 = e Z 1 + n X i =1 e X i . (76) Applying N n once more, and using ( 72 ) together with ( 69 ) – ( 71 ) , w e find N n +1 n e Z n = N n  e Z 1 + n X i =1 e X i  =  e Z 1 + n X i =1 e X i  +  e X 1 + e Z 1  + n − 1 X j =2 n X i = j +1 e X i = n X i =1 e X i + e X 1 + n X i =3 ( i − 2) e X i . W orking ov er F 2 , the co efficient ( i − 2) equals 1 if and only if i is o dd. Hence the o dd- i terms cancel against P n i =1 e X i , and w e obtain the clean form N n +1 n e Z n = X 2 ≤ i ≤ n i even e X i =: v even . (77) W e no w analyze pow ers of N n on the X -subspace span { e X 2 , . . . , e X n } . Define y k := e X n − k +1 for 1 ≤ k ≤ n − 1 , so y 1 = e X n and y n − 1 = e X 2 . F rom ( 70 ) – ( 71 ) w e hav e N n y 1 = 0 , N n y k = y 1 + y 2 + · · · + y k − 1 ( k ≥ 2) . W e claim that for every k ≥ 1 , N k − 1 n y k = y 1 and N k n y k = 0 . (78) The statement is immediate for k = 1 . F or k ≥ 2 , note that N n y k = y k − 1 + N n y k − 1 , hence N k − 1 n y k = N k − 2 n ( N n y k ) = N k − 2 n y k − 1 + N k − 1 n y k − 1 = y 1 + 0 , 24 where we used ( 78 ) for k − 1 . Applying N n once more gives N k n y k = N n y 1 = 0 , proving ( 78 ) . T aking k = n − 1 in ( 78 ) yields N n − 2 n e X 2 = e X n , N n − 1 n e X 2 = 0 . (79) Moreo v er, for an y j ≥ 3 , repeated use of ( 70 ) sho ws that N n − 2 n e X j = 0 (since the X -supp ort strictly shifts to the right at each application). Because v even in ( 77 ) con tains e X 2 and only e X j with j ≥ 3 otherwise, w e conclude N n − 2 n v even = e X n  = 0 , N n − 1 n v even = 0 . (80) Com bining ( 77 ) and ( 80 ) gives N 2 n − 1 n e Z n = N n − 2 n  N n +1 n e Z n  = N n − 2 n v even = e X n  = 0 , while N 2 n n e Z n = N n − 1 n  N n +1 n e Z n  = N n − 1 n v even = 0 . Therefore, the nilp otency index of N n is exactly 2 n . 25