Higher symmetries and logical gates of Z2 toric code

Ryohei Kobayashi
University of Maryland ( → IAS)KEK Conference 2024

Thanks to

Maissam Barkeshli(UMD)

Po-Shen Hsin
(UCLA)

Guanyu Zhu(IBM)Maissam Barkeshli, Po-Shen Hsin, RK, arXiv: 2311.05674“Higher-group symmetry of (3+1)D fermionic Z2 gauge theory: logical CCZ, CS, and T
gates from higher symmetry”

RK and Guanyu Zhu, arXiv: 2310.06917“Fault-tolerant logical gates via constant depth circuits and emergent symmetries on
non-orientable topological stabilizer and Floquet codes”

Perspective of stabilizer codes…

Shor’s 9-digit code

Toric code

Surface code

Quantum Information

Condensed Matter(Pauli) Stabilizer group

Stabilizer Hamiltonian

Code space

Ground state subspace(topological ordered phase)

Logical operators(Normalizer of S)(Emergent) symmetry of
HamiltonianHaah’s code…

High Energy
Topological Quantum
Field Theory (TQFT)

Hilbert space of TQFT

Symmetry of TQFT

Anyons…

Z2 toric code and topological order in Lab
Rather than a Hamiltonian system, it is a sequence of measurements:

Step 1: measure X stabilizers to detect Z error

correct

Step 2: measure Z stabilizers to detect X error

correct

Picture:[Fowler=Mariantoni=Martinis=Cleland]

Repeat the above process
Ø Recent experiment on realizing non-Abelian topological order (D8 gauge theory)[Iqbal et al (Quantinuum)]Logical gate of 2+1D Z2 toric code: symmetry of quantum code
Logical gates:“wrapping an anyon”

Z
X

X

X

X

Z
Z

Pauli X gate:
m particle

Pauli Z gate:
e particle

In addition, the 2d Z2 toric code (on torus) has a Z2 symmetry exchanging e

Z

X

Z

X

Z

X

m:

X

The e

m symmetry gives rise to Hadamard-like* logical gate.

(*technically it’s H1H2 SWAP)

Logical gate of 2+1D Z2 toric code(a)

X X (a)
X X

(b)

Z
Z

Z
Z

(b)

Qubits on vertices(c)Local constant-depth circuit
Permutes X and Z stabilizers:(c)em exchange

U = UB UG URClifford gates in (2+1)D by local constant-depth circuitØ Fault-tolerant logical gate is realized by a local constant-depth circuit.
Error can propagate along the shallow light cone, which is local(transversal gate = onsite symmetry)

There’s a constraint that local constant-depth circuit in 2d can only implement logical Clifford gate

[Bravyi=Koenig](for local stabilizer codes)
But Clifford gates alone cannot do universal quantum computation.
(Clifford quantum circuit can be simulated by classical computer)[Gottesman=Knill]Non-Clifford gatesØ Non-Clifford gate via local constant depth circuit: 3+1D code
Need 3d to realize fault-tolerant non-Clifford gate, which is necessary for universal QC
Clifford + a single non-Clifford = Universal.
Example: CCZ gate in 3d Z2^3 toric code. Use 0-form symmetry in (3+1)D Z2^3 gauge theory.

[Yoshida]

In 2+1D…
Ø Non-Clifford gate in 2+1D via magic state distillation
Prepare an ancilla state ‘magic state’ to implement non-Clifford gate on the code
[Bravyi=Kitaev]

Ø Non-Clifford gate in 2+1D via entangling it with 3+1D stabilizer code (code switching)[Bombin][Beverland=Kubica=Svore]

3+1D Z2 toric code with emergent fermion
Ø 3+1D Z2 toric code with emergent boson particle(3+1)D Z2 gauge theory.

(Qubits on edges of cubic lattice)

Action of effective Z2 gauge theory…

Ø 3+1D Z2 toric code with emergent fermion particle(3+1)D Z2 gauge theory with emergent fermion.

Action of effective Z2 gauge theory…

Pauli gates of 3+1D Z2 toric code w/ fermion
Quantum Information

Condensed Matter

Global symmetry

Pauli Z gate
"
!!

"
!!

"
! !

"
!!

Emergent fermion

Z2 2-form symmetry

Magnetic flux

Z2 1-form symmetry

Pauli X gate

!

!

!

!

!

What else?
Other logical gates come from generalized symmetry of Z2 gauge theory

Fault-tolerant logical gate, via pumping topological phaseØ Z2 toric code w/ emergent fermion is regarded as ‘‘Z2f gauge theory’’

Trivial atomic
insulator with Z2f
symmetry

Z2 gauge theory w/
emergent fermion

gauge Z2f symmetry(bosonization)

Symmetry defect of Z2 gauge
theorydecoration of fermionic gapped(invertible) phaseØ One can sweep the defect by (quasi-) local constant-depth circuit, which results in a logical gate.

Region wherecircuit is acted

Sweep defect byacting circuit

More symmetry of 3d Z2 toric code with emergent fermion
Ø Sweeping invertible topological phase w/ Z2f symmetry defines emergent invertible symmetry.

Sweep defectby actingcircuitZ2 (1-form) symmetry

Kitaev chain defectSweep defectby acting circuit
p+ip superconductor defectchiral phase with c- = 1/2

Z8 (0-form) symmetry[Barkeshli=Hsin=RK]

Symmetry of Z2 gauge theory in (3+1)D: when electric particle is a fermion
Action:”dynamical spin structure”

Let’s consider invertible symmetry of (3+1)D Z2 gauge theory w/ fermion:•

Z2 2-form symmetry generated by electric line operator.

•

Z2 1-form symmetry generated by magnetic surface operator.

•

Z2 1-form symmetry generated by Kitaev’s Majorana chain (1+1D spin invertible phase):on a surface•

Z8 0-form symmetry generated by gravitational CS theory (p+ip superconductor, 2+1D spin invertible phase):“p+ip symmetry defect”

0-form symmetry of (3+1)D fermionic Z2 gauge theory
Let’s look more at 0-form symmetry of (3+1)D fermionic Z2 gauge theory; why Z8?
•

Fusion rule of 0-form symmetry follows the stacking law of (2+1)D spin invertible phase, classified by
So most naively, the 0-form symmetry is Z. But some of them doesn’t act faithfully on Hilbert space.

•

Reduction from Z to Z16 happens, since 16 copies of p+ip superconductors defines bosonic E8 phase.
So, symmetry operator for 16Z is decoupled from dynamical spin structure, i.e., trivial operator on Hilbert space•

Further reduction to Z8 happens.
8 copies of p+ip superconductors does not depend on spin structure, though it needs spin structure to be defined.
i.e., trivial operator on Hilbert space.

Z4 subgroup of 0-form symmetry: 3-group symmetry
Let’s consider the Z4 subgroup of the Z8 0-form symmetry.
Z4 0-form symmetry, together with 1, 2-form symmetry, forms 3-group structure of invertible symmetries•

Z4 (subgroup) 0-form symmetry. background•Z2 1-form symmetry generated by magnetic surface. background•Z2 1-form symmetry generated by Kitaev’s Majorana chain. background•Z2 2-form symmetry generated by electric Wilson line. background3-group equation:

3-group involving Kitaev chain
1-form symmetry[Kapustin=Thorngren, Wang=Gu,
Barkeshli=Chen=Hsin=RK]

3-group involving p+ip 0-form symmetryWhole structure of 0,1,2-form symmetry becomes non-invertible
Let’s study the whole algebraic structure of the symmetry.
First, the 0-form symmetry (p+ip) induces permutation of 1-form symmetry generators:[Johnson-Freyd, Yang=Cheng]: magnetic surface operator: Kitaev chain surface operator
MZM(σ particle of Ising)Intersection between magnetic surface and p+ip defect bounds a Majorana zero mode
p+ip

Fusion rule σ x σ = 1 + ψ, σ x ψ = σ implies that total symmetry structure is non-invertible.
magnetic flux
p+ip wall
Magnetic flux’’defect-valued associator’’$1+Quantum Information

Condensed Matter

Global symmetry?

Decorate Kitaev chain,and sweep it

Z2 1-form symmetry?

Decorate p+ip superconductor,
and sweep it

Z8 0-form symmetry?

Higher-group/Non-invertible
symmetry1-form symmetry generated by Kitaev chain: logical Clifford CZ or S gate
Kitaev chainØ Kitaev chain defect:
Fermion is created at the intersection between m surface and defectmψThis effect is translated into a commutation relation of operators

Kitaev chain surface,
Clifford gate

magnetic
surface,Pauli X"

fermion line,
Pauli Z!
!

!

!

This implies that the Kitaev chain can do CZ or S like Clifford gate.
[RK=Zhu]Ø Non-trivial algebraic mixture between 1-form and 2-form symmetry: called a higher-group (3-group)

!

!

!

"
! !

"
!!

1-form symmetry generated by Kitaev chain: logical Clifford CZ or S gate
• Generic expression for logical gate (space is generic 3-manifold)

Kitaev chain
logical gate

Indices oflogical qubits: Poincare dual of
S = diag(1,i)• Kitaev chain operator can be expressed by a local constant-depth circuit, so fault-tolerant
(a)

(b)

U = UB UG UR
Qubits on vertices(c)•circuit for Kitaev chain operator

Kitaev chain circuit can also be defined on an unoriented surface: S gatee.g., Z2 toric code on a surface w/ cross-cap[RK=Zhu](FYI: Shor’s 9-digit code = Z2 toric code on real projective plane)
Shor’s codeNon-Clifford gate of 3d Z2 toric code w/ fermion: pumping a p+ip superconductor
Commutation relation of operators:… Non-Clifford gate

Ø Generic expression of p+ip logical gate on any oriented 3-manifold:: Poincare dual of

It can implement non-Clifford CCZ, Controlled-S, T gate.

Indices oflogical qubits

S = diag(1,i)
T = diag(1,exp(2πi/8))

Ø Pumping p+ip superconductor in layered system with periodic boundary condition can do CCZ gate
[Fidkowski=Hastings]
p-ip
p+ip
p-ip
p+ip
…

p-ip
p+ip

This can give rise to an exact symmetry of the (3+1)D Z2 toric code w/ fermion (after bosonization).
The unitary U is a finite time evolution by local Hamiltonian w/ exponentially decaying tails-> fault tolerance is expectedNon-Clifford gate of 3d Z2 toric code w/ fermion: pumping a p+ip superconductor
Ø Generic expression of p+ip logical gate on any oriented 3-manifold:

It can implement non-Clifford CCZ, Controlled-S, T gate.
Ø Pumping p+ip superconductor in layered system with “C2 rotation-twisted” boundary condition can do CS gate
This can make a space
(a)

(b)

yz plane is non-orientable,
so S gate…

…

Non-Clifford gate of 3d Z2 toric code w/ fermion: pumping a p+ip superconductor
Ø Generic expression of p+ip logical gate on any oriented 3-manifold:

It can implement non-Clifford CCZ, Controlled-S, T gate.
Ø It also suggests possibility of logical T gate by pumping p+ip superconductor through a 3-manifold RP3RP3 topology is realized by a ‘’partial C2 rotation’’, which can make a space(b)

non-unitary on ground state code space,
But defines non-Clifford logical gate after projecting onto code space.
On a logical qubit implemented by RP3, we have…

Quantum Information

Condensed Matter

Control-Z or S gate
(depending on choice of surface)

Decorate Kitaev chain,and sweep it

Z2 1-form symmetry

Decorate p+ip superconductor,
and sweep it

Z8 0-form symmetry

Non-Clifford CCZ, CS, or T gate
(depending on choice of 3-manifold)Commutation relation of logicals

Global symmetry

Higher-group/Non-invertible
symmetry

Future work
Ø Developing the more complete algebraic description of non-invertible symmetry of toric code
Ø Symmetry of general finite gauge theory with emergent fermionsØ Fault tolerance of T gate with partial rotation?
Ø Role of non-invertible symmetry in error correction context?
Ø New logical gates of other codes from symmetry, e.g., Floquet codes, quantum LDPC codes, etc

Backup slides

0-form symmetry of (3+1)D fermionic Z2 gauge theory
Let’s look more at 0-form symmetry of (3+1)D fermionic Z2 gauge theory; why Z8?
•

One (indirect) way to see Z8 reduction is consider (p+ip) x (p-ip), which is spin SPT phase w/ Z2 symmetry
Classified by torsion
ofThis implies that 8 copies of (p+ip) x (p-ip) doesn’t depend on spin structure of each layer.
Z2

p+ip

p-ip

Z2 background gauge field: ”difference in spin structure of two layers”

0-form symmetry of (3+1)D fermionic Z2 gauge theory
Let’s look more at 0-form symmetry of (3+1)D fermionic Z2 gauge theory; why Z8?
•

One can directly study spin-structure dependence for 8 copies of p+ip phase.
2 copies of p+ip: U(1) (spin) Chern-Simons theory: spin structure

Spin structure dependence of 2 copies:

Effect of shifting spin structure:

Spin structure dependence of 8 copies:

mod 2πZ8 reduction of 0-form symmetry v.s. mixed gravitational anomaly
The 0-form symmetry has a mixed gravitational anomaly.

This originates from framing anomaly of gravitational CS theory carrying.

One puzzle is that this is not a well-defined action of Z8 gauge field, since it’s not invariant underwhile it makes sense when C1 were Z16 gauge field.

Z8 reduction of 0-form symmetry v.s. mixed gravitational anomaly
The 0-form symmetry has a mixed gravitational anomaly.

This originates from framing anomaly of gravitational CS theory carrying.

One puzzle is that this is not a well-defined action of Z8 gauge field, since it’s not invariant underwhile it makes sense when C1 were Z16 gauge field.
Solution is to turn on background for 2-form symmetry generated by Wilson line

Invariant under the gauge transformation(

mod 2)

anomaly of Z8 symmetry is well-defined, with non-trivial mixture with 2-form symmetry: symptom of 3-groupWhole structure of 0,1,2-form symmetry becomes non-invertible
Let’s study the whole algebraic structure of the symmetry.
First, the 0-form symmetry (p+ip) induces permutation of 1-form symmetry generators:[Johnson-Freyd, Yang=Cheng]: magnetic surface
operator: Kitaev chain surface
operator

MZM(σ particle of Ising)Intersection between magnetic surface and p+ip defect bounds a Majorana zero mode,
which is a termination of the (1+1)D spin invertible phase, i.e., Kitaev’s Majorana chain•

this can be seen from compactifying Majorana fermion:

p+ip
magnetic flux

Periodic spin structure

magnetic flux

magnetic flux
compactify(2+1)D Majorana fermion with negative mass

(1+1)D Majorana fermion with negative massWhole structure of 0,1,2-form symmetry becomes non-invertible
Let’s study the whole algebraic structure of the symmetry.
First, the 0-form symmetry (p+ip) induces permutation of 1-form symmetry generators:[Johnson-Freyd, Yang=Cheng]: magnetic surface
operator: Kitaev chain surface
operator

Intersection between magnetic surface and p+ip defect bounds a Majorana zero mode,

MZM(σ particle of Ising)which is a termination of the (1+1)D spin invertible phase, i.e., Kitaev’s Majorana chain
Fusion rule σ x σ = 1 + ψ, σ x ψ = σ implies that total symmetry structure is non-invertible.
p+ip wall
Magnetic flux’’defect-valued associator’’

$
1+

p+ip
magnetic flux

Z4 subgroup of 0-form symmetry: 3-group symmetry
Let’s consider the Z4 subgroup of the Z8 0-form symmetry.
Z4 0-form symmetry, together with 1, 2-form symmetry, forms 3-group structure of invertible symmetries•

Z4 (subgroup) 0-form symmetry. background•Z2 1-form symmetry generated by magnetic surface. background•Z2 1-form symmetry generated by Kitaev’s Majorana chain. background•Z2 2-form symmetry generated by electric Wilson line. background3-group equation:

3-group involving Kitaev chain
1-form symmetry[Kapustin=Thorngren, Wang=Gu,
Barkeshli=Chen=Hsin=RK]

3-group involving p+ip 0-form symmetryZ4 subgroup of 0-form symmetry: 3-group symmetry
Let us comment on physical interpretation of the 3-group involving 0-form symmetry:Z4 0-form
magnetic

electric• second term

• first term

4 copies of p+ip

2 copies of p+ip
U(1)4

Junction of magnetic defects go through 2 copies of p+ip.

Magnetic defect intersecting with junction of 0-form symmetry

Fusion rule v x v = ψ produces an electric particle

produces a fermion (in 8 copies of p+ip, flux is fermion)1-form symmetry generated by Kitaev chain: logical Clifford CZ or S gate

Kitaev chain
logical gate•

CZ gate originates from Arf invariant on oriented surface
spin structure

torus
AP/P (= 0/1)•

AP/P (= 0/1)

Arf
1
1
1
-1

diag(1,1,1,-1) = CZ

S gate originates from Arf-Brown-Kervaire invariant on unoriented surface
8th root of unity