Fault-tolerant logical gates in quantum error-correcting codes

Transcription

1 Fault-tolerant logical gates in quantum error-correcting codes Fernando Pastawski and Beni Yoshida (Caltech) arxiv: Phys Rev A xxxxxxx Jan QIP (Sydney, Australia)

2 Fault-tolerant logical gates How do we implement a logical gate fault-tolerantly? input psi> 0> 0> 0> 0> 0> 0> Ideally, by transversal implementation U U U U U U U encoding circuit

3 The Eastin-Knill theorem (2008) Transversal logical gates are not universal for QC PRL 102, (2009) P H Y S I C A L R E V I E W L E T T E R S week ending 20 MARCH 2009 Restrictions on Transversal Encoded Quantum Gate Sets Bryan Eastin* and Emanuel Knill National Institute of Standards and Technology, Boulder, Colorado 80305, USA (Received 28 November 2008; published 18 March 2009) Transversal gates play an important role in the theory of fault-tolerant quantum computation due to their simplicity and robustness to noise. By definition, transversal operators do not couple physical subsystems within the same code block. Consequently, such operators do not spread errors within code blocks and are, therefore, fault tolerant. Nonetheless, other methods of ensuring fault tolerance are required, as it is invariably the case that some encoded gates cannot be implemented transversally. This observation has led to a long-standing conjecture that transversal encoded gate sets cannot be universal. Here we show that the ability of a quantum code to detect an arbitrary error on any single physical subsystem is incompatible with the existence of a universal, transversal encoded gate set for the code. DOI: /PhysRevLett PACS numbers: Lx, Pp Don t panic! Fault-tolerant computation is still possible.

4 The Bravyi-Koenig theorem (2012) Under a more physically realistic setting Logical gate U : low-depth unitary gate (i.e. Local unitary) Theorem For a stabilizer Hamiltonian in D dim, faulttolerantly implementable gates are restricted to the D-th level of the Clifford hierarchy. D-dim lattice???

5 Clifford hierarchy (Gottesman & Chuang) Sets of unitary transformations on N qubits P m Pauli P m = P m 1 P3 P2 P 3 Pauli P 3 = P 2 P 2 Pauli P 2 = P 1 P1 Clifford gates CNOT, Hadamard, R2 Pauli operators X,Y,Z Pauli

6 Plan of the talk Clifford hierarchy on subsystem quantum error-correcting codes Upper bound on the erasure threshold the Bravyi-Koenig theorem self-correcting quantum memory (topological order at finite temperature) Upper bound on code distance

7 Plan of the talk Clifford hierarchy on subsystem error-correcting codes Upper bound on the erasure threshold the Bravyi-Koenig theorem self-correcting memory ( at finite temperature) Upper bound on distance

8 Logical operator cleaning A logical operator can be cleaned from a correctable region. A correctable region supports no logical operator. A U B A U B correctable correctable logical operator equivalent logical operator

9 Lemma [Hierarchy] Consider a partition of the entire system into m+1 regions, denoted by R0, R1,..., Rm. If all Rj s are correctable, then transversal logical gates are restricted to m-th level Pm of the Clifford hierarchy. (eg) R1 R3 4 correctable regions R0 R2 P3

10 Consider arbitrary Pauli logical operators V0, V1,... Vm. V0 V1... Vm U0 R0, R1, R2,... Rm-1, Rm Hierarchy Pauli goal Pm U1=K(U0,V0)... U2=K(U1,V1) Um-1=K(Um-2,Vm-2)... Um=K(Um-1,Vm-1)... Pm-1 Pm-2 P1 (Pauli) Complex phase commutator : K(A,B)=ABA*B*

11 Proof of the Bravyi-Koenig theorem We can split D-dimensional system into D+1 correctable regions. (eg) 2 dim Fault-tolerant gates are in P2 *Union of spatially disjoint correctable regions = correctable region *This is not the case for subsystem codes.

12 Plan of the talk Clifford hierarchy on subsystem error-correcting codes Upper bound on the erasure threshold the Bravyi-Koenig theorem self-correcting memory ( at finite temperature) Upper bound on distance

13 Erasure Threshold Some qubits may be lost (removal errors)... eg) escape from the trap p<p loss ) Logical qubit is safe erasure threshold perror <p loss against depolarizing error

14 -operator in P (pro may mimic the one for depolarizing noise by map lizer and0 subsystem error-correcting codes with the set P3 Pauli P3 = P2 particles marked as li to the fixed-point of corr of transversally implementable logical gates. ing depolarizing channel. Hence, the loss thre ep. We assume that must necessarily be no smaller than any depola Theorem. [Loss threshold] Suppose we have a famrs supported on the threshold p p. l e ily of subsystem codes with a loss tolerance p > 1/n for onsider a transversal l perror < ploss The corollary elucidatesimplethe existing some natural number n. following Then, any transversally between mentable logical gate must loss-threshold belong to Pn and 1. the set of transvers 1 plementable gates. n Rj. We would(4)like to emphasize that the above theorem P logical gate ) p 1/n. n ` does not assume geometric locality of generators or lattheorem 2. Suppose we have a family of su e Proof sketch tice structures, and holds for arbitrary stabilizer and subcodes with a loss tolerance p > 1/n for some l Ul hassystem a tensorcodes. prod- Thenumber proof is presented in Section III. n. Then, any transversally implementab Assign each qubit to n regions uniformly at random du ]L, [U ]G denote the gate must belong to P n 1. gauge qubitsr1respec, R2,... Rn able, all the 4. dressed Subsystem codesuppose and the pcli ord hierarchy and assign each qubi All areproof. cleanable since c the regions l > 1/n, on - Rm+1, and their of n regions {Rj }j2[1,n] uniformly at random. 1 tensor prod-gates must be in Pn-1 shave a Transversal thetechnical regions chosen this way will bebk s correctable Finally, the main result is to generalize,e group commutator probability which is arbitrarily close to unity as result to subsystem codes with local generators. A diffiyerator with a tensor larger codes from the family. Finally, weto may con culty is that the so-called union lemma does not apply ersality of U and P

15 Theorem. [Loss threshold] Suppose we have a family of subsystem codes with a loss tolerance p l > 1/n for some natural number n. Then, any transversally implementable logical gate must belong to P n 1. Remarks P n logical gate ) p` 1/n. Toric code has p=1/2 threshold (related to percolation). It has a transversal P2 gate (CNOT gate) A family of codes with growing n is not fault-tolerant. Topological color code in D-dim has PD gate, so its loss threshold is less than 1/D.

16 Theorem. [Loss threshold] Suppose we have a family of subsystem codes with a loss tolerance p l > 1/n for some natural number n. Then, any transversally implementable logical gate must belong to P n 1. P n logical gate ) p` 1/n. One additional remark (due to Leonid Pryadko) Consider a stabilizer code with at most k-body generators. If the code has transversal PD logical gate, then k > O(D) D-dim color code is ~2^D body. Fewer-body code?

17 Plan of the talk Clifford hierarchy on subsystem error-correcting codes Upper bound on the loss error threshold the Bravyi-Koenig theorem self-correcting quantum memory (topological order at finite temperature) Upper bound on distance

18 Self-correcting quantum memory Can we have self-correcting memory in 3dim? Energy Does topological order exist at T>0? Energy Barrier Quantum Quantum 0> 1>

19 Theorem [Self-correction] If a stabilizer Hamiltonian in 3 dimensions has faulttolerantly implementable non-clifford gates, then the energy barrier is constant. Proof sketch Consider a partition into R0, R1, R2. Suppose that there is no string-like logical operators. Then, R0, R1, R2 are cleanable, so the code has P2 (Clifford gate) at most. String-like logical operators imply deconfined particles.

20 Theorem [Self-correction] Remark If a stabilizer Hamiltonian in 3 dimensions has faulttolerantly implementable non-clifford gates, then the energy barrier is constant. Haah s 3dim cubic code (log(l) barrier) does not have non-clifford gates. Michnicki s 3dim welded code (poly(l) barrier) does not have non-clifford gates. 6-dim color code ((4,2)-construction) has non-clifford gate and O(L) barrier. * A talk by Brell

21 Plan of the talk Clifford hierarchy on subsystem error-correcting codes Upper bound on the erasure threshold the Bravyi-Koenig theorem self-correcting memory ( at finite temperature) Upper bound on code distance

22 Theorem [Code distance] If a topological stabilizer code in D dimensions has a m-th level logical gate, then its code distance is upper bounded by 62 P m 1 d apple O(L D+1 m ). Remark Bravyi-Terhal bound for D-dim stabilizer codes (previous best) istance for top d apple O(L D 1 ) Non-Clifford gate (m>2), our bound is tighter. D-dim color code has d=l, saturating the bound.

23 Plan of the talk Clifford hierarchy on subsystem quantum error-correcting codes Upper bound on the erasure threshold the Bravyi-Koenig theorem self-correcting memory ( at finite temperature) Upper bound on distance

24 Subsystem code (generalization) Starting from non-abelian Pauli subgroup stabilizer code S = hs 1,S 2,...i H stab = X j S j subsystem code G = hg 1,G 2,...i H sub = X j G j eg) Kitaev s honeycomb model, Bacon-Shor code, gauge color code Subsystem codes require fewer-body terms. Main result For a D-dimensional subsystem code with local generators, fault-tolerantly implementable logical gates are restricted to PD if the code is fault-tolerant.

25 Breakdown of the union lemma The union lemma breaks down. A A stabilizer B dressed logical operator dressed logical operator? Non-local stabilizer operator is closely related to gapless spectrum in the Hamiltonian.

26 Proof of Bravyi-Koenig theorem We can split D-dimensional system into D+1 correctable regions. (eg) 2 dim R0 may not be correctable! (Each cycle is correctable, but union may not be correctable).

27 Fault-tolerance of the code The code must P have a finite error threshold (loss error). R2 R1 R2 R1 R1 R2 R1 R2 The union of red dots is correctable. (This circumvents the breakdown of the union lemma). R2 R1 R1 R2 R2 R1 R1 R2 Fault-tolerant logical gates are restricted to PD. In D-dimensions, fault-tolerant gates are in PD.

28 Summary of the talk Clifford hierarchy on subsystem quantum error-correcting codes Upper bound on the erasure threshold the Bravyi-Koenig theorem self-correcting quantum memory (topological order at finite temperature) Upper bound on code distance

29 Advertisement of our new paper (In)equivalence of the color code and A joint work with the toric code Aleksander Kubica Fernando Pastawski

30 Toric code vs color code? Similarities and differences between the toric code and the color code? Z Z Z Z Z Z X X X X X X Z X X X Z Z Z Z Z toric code color code * A talk by Bombin

31 aries. In particul Main results In two dimensions, we find that the triangular color code with three boundaries, defined on a lattice obtained from tiling of a sphere, is equivalent to a single copy of the toric code colored face in th on a square lattice with color boundaries, folded along a diagonal axis (see Fig.to 1). For (1) The d-dim codewhich on ais closed manifold is equivalent d > 2,multiple we find that the color code with d+1 d+1 distinct decoupled copies of boundaries the d-dimoftoric code colors up toisaequivalent local to space of aboundary. two co 1)-dimensional d copies of the transformation. toric code which are attached together at the (d unitary At this (d 1)-dimensional boundary, electric charges may condense only when electric FIG. 2. The code welding at the Extends the known result for 2dim (Yoshida2011) copies thus, t charges from all the d copies of the toric code are simultaneously present. and We remark that the construction of such a boundary may also be concisely understood by the framework of from vertices near thetoboundaries (2) The 2-dim color code with boundaries is equivalent the code welding introduced by Michnicki [? ]. folded toric code. codes by identifying the qubits co smooth A rough rough toric code folding axis B color code C On the C boundary of the to smooth not terminate with either rough

32 Main results (continued...) (3) Transversal application of Rd gates on the d-dim color code is equivalent to the generalized d-qubit control-z gate on d decoupled copies of the d-dim toric code. psi> control qubits belongs to Pd The toric code saturates the Bravyi-Koenig bound.

33 Open questions Fault-tolerant logical gates in TQFT? (eg Beverland et al 2014) The number of transversal gates? (eg Bravyi & Haah 2012) reducing the overhead of magic state distillations Non-local, but finite depth unitary? lattice rotations, lattice translations,... Many open questions, applications..., Thank you very much.