Multivariate Multicycle Codes
Stefan Krastanov University of Massachusetts AmherstUMass Quantum Group
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QuantumClifford.jl
QuantumSavory.jl
BPGates.jl
QNumerics
Good Classical Codes
rate
distance
decoder
How They Are Made
\[
H \in \mathbb{F}_2^{m\times n}
\qquad
H\ \text{sparse}
\]
Quantum Codes Need Two Codes
\[
P_X P_Z^{\!\top}=0
\]
The Random-Matrix Trap
good \(P_Z\)
compatible \(P_X\)
dense
Better Representations
expander graphs
chain complexes
Expander Graph Codes
- Panteleev, KalachevAsymptotically Good Quantum and Locally Testable Classical LDPC Codes
- Dinur et al.Good Quantum LDPC Codes with Linear Time Decoders
- Gu et al.An efficient decoder for a linear distance quantum LDPC code
Homological Algebra Codes
\[
\partial_i \partial_{i+1}=0
\]
Homological Algebra
\[
\text{boundaries have no boundary}
\]
Why Here?
surface code
lattice surgery
chain complex
Parity Checks Are Boundaries
\[
\partial(\text{error})=\text{syndrome}
\]
Surface Code
C_0X checks
\(\xleftarrow{\partial_1}\)
C_1qubits
\(\xleftarrow{\partial_2}\)
C_2Z checks
\[
P_X=\partial_1,\qquad P_Z=\partial_2^{\!\top}
\]
Divergence of Curl
\[
\partial_1\partial_2=0
\qquad\Longleftrightarrow\qquad
P_XP_Z^{\!\top}=0
\]
Really Good Quantum Codes
rate
distance
decoder
low weight
single-shot
Metachecks
\[
M_Zs_Z=0,\qquad M_Xs_X=0
\]
Complete Single-Shot Structure
K_0X metachecks
\(\xleftarrow{\partial_1}\)
K_1X checks
\(\xleftarrow{\partial_2}\)
K_2qubits
\(\xleftarrow{\partial_3}\)
K_3Z checks
\(\xleftarrow{\partial_4}\)
K_4Z metachecks
Construction
small maps
Koszul complex
candidate codes
Finite Periodic Lattice
\[
S =
\mathbb{F}_2[w,x,y,z]/
\langle w^\ell-1,\ x^m-1,\ y^p-1,\ z^r-1\rangle
\]
Local Checks
\[
F,\ G,\ H,\ I \in S
\]
Polynomial \(\to\) Circulant Matrix
\[
F(w,x,y,z)\ \mapsto\ C_F
\]
Koszul Complex
\[
K_\bullet([F,G,H,I];S)
\]
Five Spaces
SZ meta
S^4Z checks
S^6qubits
S^4X checks
SX meta
Boundary Maps
\[
\partial_4=\begin{bmatrix}F&G&H&I\end{bmatrix}
\]
\[
\partial_3=
\begin{bmatrix}
G&H&I&0&0&0\\
F&0&0&H&I&0\\
0&F&0&G&0&I\\
0&0&F&0&G&H
\end{bmatrix}
\qquad
\partial_2=
\begin{bmatrix}
H&I&0&0\\
G&0&I&0\\
0&G&H&0\\
F&0&0&I\\
0&F&0&H\\
0&0&F&G
\end{bmatrix}
\]
\[
\partial_1=\begin{bmatrix}I\\H\\G\\F\end{bmatrix}
\]
Extraction
\[
P_Z=\partial_3,\quad
P_X=\partial_2^{\!\top},\quad
M_Z=\partial_4,\quad
M_X=\partial_1^{\!\top}
\]
Consistency for Free
\[
\partial_4\partial_3=0,\qquad
\partial_3\partial_2=0,\qquad
\partial_2\partial_1=0
\]
\[
M_ZP_Z=0,\qquad
P_ZP_X^{\!\top}=0,\qquad
M_XP_X=0
\]
One Instance
\[
\begin{aligned}
F&=(1+x)(1+yz) & G&=(1+y)(1+zw)\\
H&=(1+z)(1+wx) & I&=(1+w)(1+xy)
\end{aligned}
\]
\[
[[648,60,(9,9)]]
\]
Are These Codes Good?
[[96,12,8]]
[[144,12,12]]
[[486,24,12]]
[[630,70,9]]
Confinement
\[
\min_{\operatorname{wt}(e)=w}\operatorname{wt}(Pe)
\]
Record Profiles
placeholder: confinement profile comparison
Explicit Instances
placeholder: selected MM code table
Simulation Results
Phenomenological Noise
Circuit-Level Noise
Overlapping Windows
Circuit Plateau
Takeaway
compact
single-shot
high confinement
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