Good approximate QLDPC codes from spacetime Hamiltonians Chinmay - PowerPoint PPT Presentation

Good approximate QLDPC codes from spacetime Hamiltonians Chinmay Nirkhe joint work with Thom Bohdanowicz Elizabeth Crosson Henry Yuen Caltech University of New Mexico University of Toronto arXiv:1811.00277 QIP 2019 nirkhe@cs.berkeley.edu

Why study error-correcting codes? Quantum fault tolerance Quantum Witness [Gottesman 09 ] QLDPC ⇒ fault tolerance quantum • computation with constant overhead Quantum PCP conjecture input 𝑦 Verifier Hardness of approximation in quantum setting • Entanglement at room temperature • Interesting local Hamiltonians with robust entanglement properties • toric code, color codes, etc. • Image: Daniel Gottesman, APS Chinmay Nirkhe Approx. QLDPC codes from spacetime Hamiltonians

What makes a code good? Rate Distance Enc |𝜒⟩ |𝜒⟩ Decoding Circuit Encoding Circuit Enc |𝜒⟩ |𝜒⟩ Stabilizer weight (locality) Rate: 𝑙 𝑜 = Ω(1) Distance: 𝑒 = Ω(𝑜) Locality: 𝑃(1) 𝐼 3 … 𝐼 1 𝐼 2 Chinmay Nirkhe Approx. QLDPC codes from spacetime Hamiltonians

We show that optimal rate, distance and locality parameters are possible (modulo polylog corrections) if we go beyond stabilizer codes to non-commuting and approximate codes Chinmay Nirkhe Approx. QLDPC codes from spacetime Hamiltonians

Quantum error Local correcting codes Hamiltonians ex. toric code (with robust entanglement) this talk Chinmay Nirkhe Approx. QLDPC codes from spacetime Hamiltonians

Outline • Coding theory definitions • Uniformization via sorting circuits • Spacetime Hamiltonians • Spectral gap analysis Chinmay Nirkhe Approx. QLDPC codes from spacetime Hamiltonians

What is a LDPC code? Classically, a code 𝒟 is a dim 𝑙 subspace of ℤ 2 𝑜 . 𝑜× 𝑜−𝑙 . A linear code can be defined by a matrix 𝐼 ∈ ℤ 2 ow L 𝑜 ∶ 𝐼𝑦 = 0 ensity D 𝒟 = 𝑦 ∈ ℤ 2 𝑦 1 arity P 1 1 0 1 1 0 𝑦 2 𝐼 = = 0 heck C 0 1 1 0 1 1 𝑦 3 𝑦 1 = 𝑦 2 = 𝑦 3 ⇒ 𝒟 = {000, 111} 𝐼 has 𝑑 -locality if 𝐼 is 𝑑 -row sparse and 𝑑 -column sparse. Chinmay Nirkhe Approx. QLDPC codes from spacetime Hamiltonians

Benefits of an LDPC code Since the checks overlap, they 1 1 0 𝐼 = can’t be parallelized and must 0 1 1 be done in series. 0 If the code is 𝑑 -local, then the 1 checks can be parallelized into 1 𝑑 3 + 𝑑 depth circuit. 1 0 Proof: Each check shares bits with at most 𝑑 2 other checks. By coloring argument, requires 𝑑 2 + 1 rounds. Each round requires depth 𝑑 . Chinmay Nirkhe Approx. QLDPC codes from spacetime Hamiltonians

Quantum LDPC codes 𝑎 For CSS codes 𝑌 𝑎 𝑎 𝑎 (codes that handle 𝑌 𝑌 𝑌 errors and 𝑎 𝑌 errors separately), definition is easy… both parity check ex. toric code matrices 𝐼 𝑌 and 𝐼 𝑎 rate: 2/𝑜 locality: 4 need to have low density. Can we do better? distance: 𝑃( 𝑜) Chinmay Nirkhe Approx. QLDPC codes from spacetime Hamiltonians

Best known stabilizer codes • [Tillich-Zemor 13 ] • rate: Ω(1) • distance: 𝑃( 𝑜) • locality: 𝑃(1) To do better, we probably need • [Freedman-Meyer-Luo 02 ] to go past stabilizer codes! • rate: Ω(1/𝑜) • distance: 𝑃( 𝑜 log 𝑜) • locality: 𝑃(1) Chinmay Nirkhe Approx. QLDPC codes from spacetime Hamiltonians

Going past stabilizer codes Let 𝐼 1 , 𝐼 2 , … , 𝐼 𝑛 be a set of 𝑑 -local not necessarily commuting projectors acting on 𝑜 qubits. Define the code-space 𝒟 as the mutual eigenspace: 𝜒 ∈ ℂ 2 ⊗𝑜 𝜒 𝐼 𝑗 𝜒 = 0 ∀ 𝐼 𝑗 𝒟 = 𝐼 = 𝐼 1 + ⋯ + 𝐼 𝑛 is 𝑑 -QLDPC if additionally each qubit participates in at most 𝑑 terms 𝐼 𝑗 . Chinmay Nirkhe Approx. QLDPC codes from spacetime Hamiltonians

First attempt [N-Vazirani-Yuen 18 ] CSS codes exist with linear rate and distance, but lack locality. Create a Hamiltonian whose ground-space is almost exactly that of a CSS code but is locally checkable. Chinmay Nirkhe Approx. QLDPC codes from spacetime Hamiltonians

First attempt [N-Vazirani-Yuen 18 ] Create a Hamiltonian whose ground-space is almost exactly that of a CSS code but is locally checkable. Express a computation as the ground-state of a 5-local Hamiltonian (Feynman-Kitaev clock Hamiltonian) [Kitaev 99 ] Together, {|𝜔 𝑢 ⟩} are a “proof” that the circuit was executed correctly. But, ෩ Ψ = 𝜔 0 𝜔 1 … |𝜔 𝑈 ⟩ is not 𝐵 locally-checkable. 𝜔 0 = 𝜊 0 |𝜊⟩ 𝜔 1 = 𝐵 𝜔 0 𝐷 Instead, the following ”clock” state* is: 𝜔 2 = 𝐶|𝜔 1 ⟩ 𝑈 𝐶 1 𝜔 3 = 𝐷|𝜔 2 ⟩ |0⟩ Ψ = ෍ 𝑢 |𝜔 𝑢 ⟩ 𝑈 + 1 𝑢=0 |𝜔 0 ⟩ |𝜔 1 ⟩ |𝜔 2 ⟩ |𝜔 3 ⟩ *Quantum analog of Cook 71 -Levin 73 Tableau. Chinmay Nirkhe Approx. QLDPC codes from spacetime Hamiltonians

First attempt [N-Vazirani-Yuen 18 ] Create a Hamiltonian whose ground-space is almost exactly that of a CSS code but is locally checkable. Let 𝐷 = 𝐷 𝑈 𝐷 𝑈−1 … 𝐷 1 be a circuit with gates {𝐷 𝑗 } and let 𝜔 0 = |𝜊⟩|0⟩ ⊗𝑜−𝑙 be an initial state for |𝜊⟩ ∈ ℂ 2 ⊗𝑙 . There is a local Hamiltonian with ground space of: 𝑈 1 𝜔 𝑢 = 𝐷 𝑢 𝜔 𝑢−1 , 𝒣 = ቐ Ψ 𝜊 = ෍ unary 𝑢 ⊗ 𝜔 𝑢 ∶ ቑ . 𝜔 0 = |𝜊⟩|0⟩ ⊗(𝑜−𝑙) 𝑈 + 1 𝑢=0 Chinmay Nirkhe Approx. QLDPC codes from spacetime Hamiltonians

First attempt [N-Vazirani-Yuen 18 ] Create a Hamiltonian whose ground-space is almost exactly that of a CSS code but is locally checkable. Let 𝑊 be the encoding 𝐷 𝑊 𝕁 circuit for a good CSS = code. 𝐿 long Choose 𝐿 = 𝑃 𝑈 𝑊 𝜀 −2 . Construct the clock Hamiltonian for this “padded” circuit 𝐷 . Chinmay Nirkhe Approx. QLDPC codes from spacetime Hamiltonians

First attempt [N-Vazirani-Yuen 18 ] The groundspace of 𝐼 is ≈ the groundspace of a CSS code tensored with junk. 𝑈 1 : 𝜔 𝑢 = 𝐷 𝑢 𝐷 𝑢−1 … 𝐷 1 𝜔 0 , 𝒣 𝐷 = ቐ ෍ 𝑢 𝜔 𝑢 ቑ 𝜔 0 = |𝜊⟩|0⟩ ⊗(𝑜−𝑙) 𝑈 𝐷 + 1 𝑢=0 But for 𝑢 ≥ 𝑈 𝑊 , 𝜔 𝑢 = 𝑊 𝜔 0 . Thus, 1 − 𝑃(𝜀 2 ) fraction of 𝜔 𝑢 = 𝑊 𝜔 0 . 𝑈 1 𝑊 𝜔 0 ∶ 𝜔 0 = 𝜊 0 ⊗(𝑜−𝑙) . 𝒣 𝐷 ≈ ෍ 𝑢 ⊗ ൛ ൟ 𝑈 𝐷 + 1 𝑢=0 Plus, 𝒣 𝐷 is the ground-space of a However, some qubits 5-local Hamiltonian! participate in many terms 𝐼 𝑢 . Chinmay Nirkhe Approx. QLDPC codes from spacetime Hamiltonians

First attempt [N-Vazirani-Yuen 18 ] However, some qubits participate in many terms 𝐼 𝑢 . 𝐼 𝑢 checks that the slice 𝑢 |𝜔 𝑢 ⟩ and the slice 𝑢 + 1 𝜔 𝑢+1 satisfy 𝜔 𝑢+1 = 𝑉 𝑢 |𝜔 𝑢 ⟩ 𝑢 th gate of circuit Locality of the code corresponds to the connectivity of the qubits in the circuit. Minimize connectivity of the qubits in the circuit. Chinmay Nirkhe Approx. QLDPC codes from spacetime Hamiltonians

Localizing the circuit via bitonic sorting circuits Minimize connectivity of the qubits in the circuit. Theorem [Batcher 65 ]: There is a circuit of depth log 2 𝑜 with log 𝑜 connectivity sorting 𝑜 elements. Can stretch circuit by log 2 𝑜 mult. depth and reduce connectivity to 𝑜 . Can be used anywhere to simplify circuit connectivity in any situation. Chinmay Nirkhe Approx. QLDPC codes from spacetime Hamiltonians

Long clocks and brittle Hamiltonians For Feynman-Kitaev clock Hamiltonian each layer of the circuit needs exactly 1 gate. 1 2 2 1 3 4 This yields long clocks and brittle Hamiltonians. Brittle Hamiltonian: Small spectral gap. Not satisfying any 1equation of 𝜔 𝑢+1 = 𝑉 𝑢 |𝜔 𝑢 ⟩ has energy 𝑃(1/|𝐷|) with |𝐷| = the number of gates in 𝐷 . Chinmay Nirkhe Approx. QLDPC codes from spacetime Hamiltonians

Long clocks and brittle Hamiltonians This yields long clocks and brittle Hamiltonians. Build Hamiltonian with ground-state of There are more than |𝐷| partial computations of a uniform superposition circuit! overall partial = 2 2 1 |1⟩ computations 𝜐 : B ෍ 𝜐 |𝜔 𝜐 ⟩ A C B D A D B C Space-time C Hamiltonian [Breuckmann-Terhal 14 ] Chinmay Nirkhe Approx. QLDPC codes from spacetime Hamiltonians

Long clocks and brittle Hamiltonians This yields long clocks and brittle Hamiltonians. Build Hamiltonian with Theorem: This yields a Hamiltonian for whom the spectral gap scales ground-state of There are more than |𝐷| partial computations of a 1 uniform superposition ෨ 𝑃 𝑜 3.09 depth 𝐷 2 circuit! overall partial = 2 2 1 |1⟩ 1 Instead of 𝑃 as in standard Feynman-Kitaev clock Hamiltonian computations 𝜐 : 𝐷 B ෍ 𝜐 |𝜔 𝜐 ⟩ A C B D A D B C Space-time C Hamiltonian [Breuckmann-Terhal 14 ] Chinmay Nirkhe Approx. QLDPC codes from spacetime Hamiltonians