Topological Quantum Computation

School US-Japan seminar 2013/4/4 @Nara

Topological quantum computation -from topological order to fault-tolerant quantum computation- The Hakubi Center for Advanced Research, Kyoto University Graduate School of Informatics, Kyoto University Keisuke Fujii Outline

(1) Introduction: what is topological order? condensed matter physics (2) Majorana fermions & 2D Kitaev model

(3) Thermal instability of topological order

(4) Error correction on (Kitaevʼs toric code) surface code

(5) Topological quantum computation quantum defect qubits/ braiding /magic state information distillation/ implementations processing What is topological order?

Topological order is...... -a new kind of order in zero-temperature phase of matter.

-cannot be described by Landauʼs symmetry breaking argument.

-ground states are degenerated and it exhibits long-range quantum entanglement.

-the degenerated ground states cannot be distinguished by local operations.

-topologically ordered states are robust against local perturbations.

-related to quantum spin liquids, fractional quantum Hall effect, fault-tolerant quantum computation. Landau’s symmetry breaking argument

Ising model (e.g. two dimension): Pauli operators: H = J X X − i j 10 01ij 0 i 10 I = ,X = ,Y = − ,Z = . the 01Ising Hamiltonian10 is invariant spin ﬂippingi 0 Z w.r.t. X-basis.0 1 −

ComputationalMagnetization (local basis order (qubit): parameter) takes zero above the critical temperature (CT) and non-zero below the CT. 0 , 1 → Z 0 = 0 ,Z1 = 1 {| | } | | | −| + , → X + = + ,X = {| |−} | | |− −|− disordered N (para) Multi-qubit system 0 , 1 ⊗ : {| | } Ai = I1 ... Ii 1orderedAi Ii+1 ...IN −

⊗ ⊗magnetization (ferro)⊗ ⊗ ⊗ (A = X, Y, Z) temperature T Landau’s symmetry breaking argument

Ising model (e.g. two dimension): H = J X X − i j ij The Ising Hamiltonian is invariant under spin ﬂipping Z w.r.t. X-basis.

The ground states: ++ + , (Z2 symmetry) {| ··· | −−···−} ... longitudinal magnetic ﬁeld | ↑↑ ↑

gs h X 2Nh ++ + i | ··· | −−···− ... | ↓↓ ↓ →ground state degeneracy is lifted by longitudinal magnetic ﬁeld.

→ground state degeneracy is not robust against local perturbation. Topologically ordered states

In topologically ordered system....

degenerated ground states: any local perturbation ΨGS1 Ψ | | GS2 well separated

O(L) Only nonlocal operators (high weight operator A ⊗ ) can exchange the ground states. → The ground state degeneracy cannot be lifted by any local αL operations (energy shift is ~ e − ). e.g) Ψ =(0000 + 1111 )/√2 | 1 | | Ψ =(0011 + 1100 )/√2 | 2 | | two orthogonal state cannot be distinguished by measuring a single qubit in any basis → second order perturbation ﬁrst lifts the degeneracy Topologically ordered states

How can we describe topologically ordered states efﬁciently? → Theory of quantum error correction is a very useful tool to describe topologically ordered system. quantum error correction codes topologically ordered system

code subspace ground state degeneracy correctability against errors robustness against local perturbation (k-error correction code) (robust up to (2k+1)-th order perturbation) stabilizer codes stabilizer Hamiltonian (D. Gottesman PhD thesis 97) locality and translation invariance Toric code (surface code) Kitaev model classical repetition code Ising model (non-topological-ordered) (can correct either X or Z errors) → thermal stability/ information capacity of discrete systems/ exotic topologically ordered state (fractal quantum liquid) Outline

(1) Introduction: what is topological order? condensed matter physics (2) Majorana fermions & 2D Kitaev model

(3) Thermal instability of topological order

(4) Error correction on (Kitaevʼs toric code) surface code

(5) Topological quantum computation quantum defect qubits/ braiding /magic state information distillation/ implementations processing Majorana fermions

Ising model in one dimension (Ising chain): H = J X X − i j ij the Ising Hamiltonian is invariant spin ﬂipping Z w.r.t. X-basis.

The ground states: ++ + , (Z2 symmetry) {| ··· | −−···−}

longitudinal magnetic ﬁeld

gs h X 2Nh ++ + i | ··· | −−···−

a repetition code X i X i +1 cannot correct any X error, since any single bit-ﬂip error X i changes the code space non-trivially. Majorana fermions

Let us consider a mathematically equivalent but physically different system.

N 1 − Ising chain: H = J X X Ising − i i+1 i=1

Jordan-Wigner transformation c2i 1 = Z1 ...Zi 1Xi ⇄ − − (spin fermion) c2i = Z1 ...Zi 1Yi − ci,cj = δijI,ci† = ci {(Majorana} fermion operator) N 1 − 2N spinless fermions: H = J ( i)c c Maj − − 2i 2i+1 2 superconductor, topological insulator, semiconducting heterostructure (see A. Kitaev and C. Laumann, arXiv:0904.2771 for review ) Majorana fermions

N 1 − paired H = J ( i)c c ・・・ Maj − − 2i 2i+1 2 c1 c2 c3 c4 c2N 1 c2N − ground states: ( i)c c Ψ = Ψ for all i. − 2i 2i+1| | unpaired Majorana fermions at the edges of the chain → “zero-energy Majorana boundary mode” 0¯ , 1¯ {| | } ( i)c c 0¯ = 0¯ , ( i)c c 1¯ = 1¯ ,c0¯ = 1¯ − 1 2N | | − 1 2N | −| 1| |

Y1Z2 ...ZN 1YN (Z2 symmetry) X1 − If unpaired Majonara fermions are well (act on the ground subspace nontrivially) separated, this operator would not act.

But! c 1 or c 2 N (odd weight fermionic operators) require coherent creation/ annihilation of a single fermion, which is prohibited by superselection rule. → X errors are naturally prohibited by the fermionic superselection rule. → Unpaired Majorana fermion is robust against any “physical” perturbation. Topological quantum computation

Majorana fermions are non- Abelian, but do not allow universal quantum computation.

Topologically protected gates + Magic state distillation [Bravyi-Kitaev PRA 71, 022316 (2005)] =

Universal quantum computation pair creation of exchanging Majorana Majorana fermions fermions via T-junction Topological quantum computation

Majorana fermions are non- Abelian, but do not allow universal quantum computation.

Topologically protected gates What if your system has no + superselection rule, suchMagic asstate the distillation [Bravyi-Kitaev PRA 71, 022316 (2005)]

fermionic parity preservation?=

Universal quantum computation pair creation of exchanging Majorana Majorana fermions fermions via T-junction Kitaev’s honeycomb model A. Kitaev, Ann. Phys. 321, 2 (2006)

Honeycomb model: H = J X X J Y Y J Z Z hc − x i j − y i j − z i j x-link y-link z-link x y x y z z z up y x y x dimerization z z z z left right y x y x y x Jx,Jy Jz z z z down y x y x J 2J 2 H = x y Y Y X X eﬀ −16 J 3 left(p) right(p) up(p) down(p) z p | | Toric code Hamiltonian: local unitary H = J Z Z Z Z J X X X X transformation TC − l(f) r(f) d(f) u(f) − l(v) r(v) d(v) u(v) f v A. Kitaev, Ann. Phys. 303, 2 (2003) Kitaev’s toric code model

Kitaevʼs toric code model is a representative example of topologically ordered system.

face operator: Af = Zi Z i face f X ∈ Z Af Z X vertex operator: Bv = Xi Bv Z i vertex v X X ∈ Toric code Hamiltonian: H = J A J B − f − v v f Note that these operators are commutale:

even number anti-commute × anti-commute crossover ＝ commute The ground states are given by simultaneous eigenstate of all face & vertex operators (gapped and frustration-free): A Ψ = Ψ ,BΨ = Ψ f | | v| | Kitaev toric code model

Kitaevʼs toric code model is a representative example of topologically ordered system. ★Short note on the stabilizer formalism

face operator: Af =n Zi •n-qubit Pauli group: 1, i I,X,Y,Z ⊗ Z {± ± } × { } i face f X ∈ Z•stabilizerAf Z group: = vertexSi , where operator: [Si,Sj] = 0 and Si = S† X S { } Bv = iXi Bv (commutative) (hermitian) Z i vertex v X X ∈ stabilizer generators: minimum independent set of stabilizer elements • Toric code Hamiltonian: H = J A J B − f − v v f •stabilizer state: S i Ψ = Ψ for all stabilizer generators S i |Note that| these operators are commutale:

•example: X1X2,Z1Z2 ( 00 + 11 )/√2 even→ number| | anti-commute × anti-commute crossing ＝ commute dimension of the stabilizer subspace: 2^(# qubits - # generators) • The ground states are given by simultaneous eigenstate of all face & vertex operators (gapped and frustration-free): A Ψ = Ψ ,BΨ = Ψ f | | v| | Kitaev toric code model

Kitaevʼs toric code model is a representative example of topologically ordered system.

face operator: Af = Zi Z i face f X ∈ →stabilizer generators Z Af Z X vertex operator: Bv = Xi Bv Z i vertex v X X ∈ Toric code Hamiltonian: H = J A J B − f − v v →stabilizer Hamiltonian f Note that these operators are commutale:

even number anti-commute × anti-commute crossing ＝ commute The ground states are given by simultaneous eigenstate of all face & vertex operators (gapped and frustration-free): →stabilizer subspace A Ψ = Ψ ,BΨ = Ψ f | | v| | Structure of grand states

Degeneracy of the ground subspace: Z X [Torus] Z Af Z 2 X # qubits: (edges) on N×N torus = 2N Bv Z X # stabilizer generators: X 2 (faces + vertexes - 2) = 2N - 2

# dimension of ground subspace: 22=4 # logical qubits → 2 (two logical qubits)

[General surface] (face)+(vertex)-(edge)=2-2g g = genus Euler characteristic

# logical qubits → (edge)-[(face)+(vertex)-2]=2g Structure of ground states

Degeneracy of the ground subspace: Z X [Torus] Z Af Z 2 X # qubits: (edges) in N×N torus = 2N Bv Z X # stabilizer generators: X 2 (faces + vertexes - 2) = 2N - 2

# dimension of ground subspace: 22=4 # logical qubits → 2 (two logical qubits)

[General surface] How is the ground (face)+(vertex)-(edge)=2-2state degeneracy described?g g = genus Euler characteristic → Find a good quantum number! The operator that acts on the ground subspace #nontrivially, logical qubits “logical→ (edge)-[(face)+(vertex)-2]=2 operator”. g Non-trivial cycle: Logical operators

L X(¯c1 )

L L The operators on non-trivial cycles Z ( c 1 ) ,X (¯ c 1 ) are commutable with all face and vertex operators, but L cannot given by a product of them. X(¯c1 ) L L L Z(c ),X(¯c ) =0 Z(c1 ) { 1 1 } → logical Pauli operators.

g=1 → # of logical qubit = 2:

L L L L L Z(c1 ),X(¯c1 ) , Z(c1 ),X(¯c1 ) Z(c1 ) { } { } (The action of logical operators depend only on the homology class of the cycle.)

The logical operators have weight N. → N-th order perturbation shifts the ground energy. Stability against local perturbations

local ﬁeld terms

H = HTC + hx Xi + hz Zi i i Z2 Ising gauge model quantum/classical mapping (dual of 3D Ising model) by Trotter-Suzuki expansion

topologically ordered (Higgs phase)

Tupitsyn et al., PRB 82, 085114 (2010) Stability against local perturbations

local ﬁeld terms

H = HTC + hx Xi + hz Zi i i Z2 Ising gauge model quantum/classical mapping (dual of 3D Ising model) by Trotter-Suzuki expansion Is stability against perturbations enough for fault-tolerance? No. Stability against thermal topologically ordered (Higgs phase) ﬂuctuation is also important!

Tupitsyn et al., PRB 82, 085114 (2010) Outline

(1) Introduction: what is topological order? condensed matter physics (2) Majorana fermions & 2D Kitaev model

(3) Thermal instability of topological order

(4) Error correction on (Kitaevʼs toric code) surface code

(5) Topological quantum computation quantum defect qubits/ braiding /magic state information distillation/ implementations processing Thermal instability of topological order

Majorana fermion: ・・・ c1 c c3 c4 c2N 1 c2N 2 −

++++ + + ++ ++ + ++ + + 1st| − | −−−− ・・・ | −−−−−−− excitation domain growth Zi = c2i 1c2i − gs + + + + + + + ++ | | − − − − − − − −−

Excitation (domain-wall) is a point-like object. ・・・

→There is no large energy barrier between the degenerated ground states.

gs Thermal instability of topological order

Kitaevʼs toric code model:

anyonic excitation Anyon can move freely (Abelian) X X X X without any energetic penalty. →excitation is a point-like X object. X

pair creation pair annhilation

1st ・・・ excitation domain growth gs Thermal instability of topological order

More generally... Topological order in any local and translation invariant stabilizer Hamiltonian systems in 2D and 3D do not have thermal stability. quantum error 2D: S. Bravyi and B. Terhal, New J. Phys. 11, 043029 (2009). correction 3D: B. Yoshida, Ann. Phys. 326, 2566 (2011). code theory Thermally stable topological order (self-correcting quantum memory) in 4D by E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, J.Math.Phys. 43, 4452 (2002). (Excitation has to be two-dimensional object for each non- commuting errors, X and Z. →4D)

Existence/non-existence of thermally stable topological order (= self-correcting quantum memory) in 3 or lower dimensions is one of the open problems in physics! (see list of unsolved problem in physics in wiki)

Non-equilibrium condition (feedback operations) is necessary to observe long-live topological order (many-body quantum coherence) at ﬁnite temperature. Outline

(1) Introduction: what is topological order? condensed matter physics (2) Majorana fermions & 2D Kitaev model

(3) Thermal instability of topological order

(4) Error correction on (Kitaevʼs toric code) surface code

(5) Topological quantum computation quantum defect qubits/ braiding /magic state information distillation/ implementations processing Topological error correction

face stabilizer: Af = Zi Z i face f X ∈ Z Af Z X vertex stabilizer: Bv = Xi Bv Z i vertex v X X ∈ Toric code Hamiltonian: H = J A J B − f − v v f The code state is deﬁed by A Ψ = Ψ ,BΨ = Ψ f | | v| | for all face and vertex stabilizers. Errors on the surface code

If a chain of X (bit-ﬂip) errors occurs, the eigenvalues of the face stabilizers become -1 at the boundary of the error chain.

(In the toric code Hamiltonian, they correspond to the anyonic excitations) Errors on the surface code

Similarly if a Z (phase-ﬂip) error chain occurs, the eigenvalues of the vertex stabilizers become -1 at boundary of the error chian.

(that is, toric code model have two types of anyonic excitations) For simplicity, we only consider X errors correction below. Errors on the surface code

If a chain of X (bit-ﬂip) errors occurs, the eigenvalues of the face stabilizers become -1 at the boundary of the error chain.

(In the toric code Hamiltonian, they correspond to the anyonic excitations) Syndrome measurements

Measure the eigenvalues of the stabilizer operators. Projective measurement for an operator A （hermitian & eigenvalues ±1） + | X 0 0 I + 1 1 A Af | | ⊗ | | ⊗ ψ Bv | A

X X

+ | + | (In the toric code Hamiltonian, the syndrome measurements correspond to measurements of the local energy.) Topological error correction

The syndrome measurements do not tell us the actual location of errors, but boundaries of them. (It tells location of excitations, but does not tell the trajectory of the excitaitons)

Then we have to infer a recovery chain, to recover from errors. In the toric code Hamiltonian, this can be viewed as ﬁnding an appropriate way to annihilate pairs of anyones. Topological error correction

If error and recovery chains result in a trivial cycle, the error correction succeeds.

Trivial cycle = stabilizer element Actual and estimated error locations are the same. Topological error correction

If the estimation of the recovery chain is bad .... Topological error correction

If the estimation of the recovery chain is bad ....

The error and recovery chains result in a non-trivial cycle, which change the code state. Algorithms for error correction

syndrome recovery chain

→ The error chain which has the highest probability conditioned on the error syndrome. → minimum-weight-perfect-match (MWPM) algorithm (polynomial algorithm) Blossom 5 by V. Kolmogorov, Math. Prog. Comp. 1, 43 (2009). [Improved algorithms] by Duclos-Cianci & Poulin Phys. Rev. Lett. 104, 050504 (2010). by Fowler et al., Phys. Rev. Lett. 108, 180501 (2012). by Wootton & Loss, Phys. Rev. Lett. 109, 160503 (2012).

E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, J.Math. Phys. 43, 4452 (2002). Algorithm for error correction

p=3% 0.8 30 0.7 25 — N=10 0.6 — N=20 20 0.5 — N=30 15 0.4 p=10.3% 10 0.3 5 0.2 threshold

0 0.1 value 0 5 10 15 20 25 30 logical error probability error logical 0 p=10% p=15% 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 3030 3030 physical error probability 2525 2525

2020 2020 The inference problem can

1515 1515 be mapped to a ferro-para phase transition of random- 10 1010 10 bond Ising model. 55 55 E. Dennis, A. Kitaev, A. Landahl, 00 00 and J. Preskill, J.Math. Phys. 43, 00 55 1010 15 20 25 30 00 55 1010 1515 2020 2525 3030 4452 (2002). Noise model and threshold values

Code performance: X Independent X and Z Dennis et al., errors with perfect syndrome J. Math. Phys. 49, 4452 (2002). measurements. M. Ohzeki, + Phys. Rev. E 79 021129 (2009). [10.3-10.9%] |

Phenomenological noise model: X Independent X and Z Wang-Harrington-Preskill, errors with noisy syndrome Ann. Phys. 303, 31 (2003). measurements. Ohno et al., + [2.9-3.3%] | Nuc. Phys. B 697, 462 (2004).

Circuit noise model: X Raussendorf-Harrington-Goyal, Errors are introduced by NJP 9, 199 (2007). each elementary gate. Raussendorf-Harrington-Goyal, Ann. Phys. 321, 2242 (2006). [0.75%] + | Outline

(1) Introduction: what is topological order? condensed matter physics (2) Majorana fermions & 2D Kitaev model

(3) Thermal instability of topological order

(4) Error correction on (Kitaevʼs toric code) surface code

(5) Topological quantum computation quantum defect qubits/ braiding /magic state information distillation/ implementations processing p- and d-type defects

too complex....

too complex.... introduce “defects” on the (defect = removal of the stabilizer planer surface code operator from the stabilizer group, which introduce a degree of freedom)

primal defect pair

dual defect pair dynamics of defects

• preparation of logical qubit →creation of defect pair

• moving the defect

• measurement of logical qubit pair annihilation of defects

• braiding p-defect around d-defect →Controlled-Not gate between p-type (control) and d-type (target) qubits.

p-type qubit = d-type qubit Prepare & move the defect

Preparation of eigenstate + p of L p : | L X

X X X X X Aa Ab Aa Ac Aa Ac

X basis measurement X basis measurement stabilizer measurement Moving the defect：

logical operator repeat surface code p expand shrink primalmove the defect+ defect pair | L primal defect pair creation time CNOT gate by braiding