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

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

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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

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SLIDE 1

Good approximate QLDPC codes from spacetime Hamiltonians

Chinmay Nirkhe

joint work with nirkhe@cs.berkeley.edu Thom Bohdanowicz Caltech Elizabeth Crosson University of New Mexico Henry Yuen University of Toronto arXiv:1811.00277 QIP 2019

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SLIDE 2

Chinmay Nirkhe

  • Approx. QLDPC codes from spacetime Hamiltonians

Why study error-correcting codes?

Quantum fault tolerance

  • [Gottesman09] QLDPC β‡’ fault tolerance quantum

computation with constant overhead

Interesting local Hamiltonians

  • with robust entanglement properties
  • toric code, color codes, etc.

Quantum PCP conjecture

  • Hardness of approximation in quantum setting
  • Entanglement at room temperature

Image: Daniel Gottesman, APS

input 𝑦 Verifier Quantum Witness

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SLIDE 3

Chinmay Nirkhe

  • Approx. QLDPC codes from spacetime Hamiltonians

What makes a code good?

Rate

|πœ’βŸ© Enc |πœ’βŸ©

Distance

Enc |πœ’βŸ© |πœ’βŸ© Decoding Circuit Encoding Circuit

Stabilizer weight (locality)

𝐼1 𝐼2 𝐼3 …

Rate: 𝑙

π‘œ = Ξ©(1)

Distance: 𝑒 = Ξ©(π‘œ) Locality: 𝑃(1)

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SLIDE 4

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

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SLIDE 5

Chinmay Nirkhe

  • Approx. QLDPC codes from spacetime Hamiltonians

Quantum error correcting codes Local Hamiltonians

(with robust entanglement)

  • ex. toric code

this talk

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SLIDE 6

Outline

  • Coding theory definitions
  • Uniformization via sorting circuits
  • Spacetime Hamiltonians
  • Spectral gap analysis

Chinmay Nirkhe

  • Approx. QLDPC codes from spacetime Hamiltonians
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SLIDE 7

Chinmay Nirkhe

  • Approx. QLDPC codes from spacetime Hamiltonians

What is a LDPC code?

  • w

ensity arity heck L D P C

Classically, a code π’Ÿ is a dim 𝑙 subspace of β„€2

π‘œ.

A linear code can be defined by a matrix 𝐼 ∈ β„€2

π‘œΓ— π‘œβˆ’π‘™ .

π’Ÿ = 𝑦 ∈ β„€2

π‘œ ∢ 𝐼𝑦 = 0

1 1 1 1 𝑦1 𝑦2 𝑦3 = 0 𝐼 = 1 1 1 1 𝑦1 = 𝑦2 = 𝑦3 β‡’ π’Ÿ = {000, 111} 𝐼 has 𝑑-locality if 𝐼 is 𝑑-row sparse and 𝑑-column sparse.

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Chinmay Nirkhe

  • Approx. QLDPC codes from spacetime Hamiltonians

Benefits of an LDPC code

𝐼 = 1 1 1 1

1 1

Since the checks overlap, they can’t be parallelized and must be done in series. If the code is 𝑑-local, then the checks can be parallelized into 𝑑3 + 𝑑 depth circuit.

1 Proof: Each check shares bits with at most 𝑑2 other checks. By coloring argument, requires 𝑑2 + 1 rounds. Each round requires depth 𝑑.

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SLIDE 9

Chinmay Nirkhe

  • Approx. QLDPC codes from spacetime Hamiltonians

Quantum LDPC codes

For CSS codes (codes that handle π‘Œ errors and π‘Ž errors separately), definition is easy… both parity check matrices πΌπ‘Œ and πΌπ‘Ž need to have low density.

π‘Œ π‘Œ π‘Œ π‘Œ π‘Ž π‘Ž π‘Ž π‘Ž

locality: 4 distance: 𝑃( π‘œ) rate: 2/π‘œ

Can we do better?

  • ex. toric code
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SLIDE 10

Best known stabilizer codes

  • [Tillich-Zemor13]
  • rate: Ξ©(1)
  • distance:𝑃( π‘œ)
  • locality:𝑃(1)
  • [Freedman-Meyer-Luo02]
  • rate: Ξ©(1/π‘œ)
  • distance:𝑃( π‘œ log π‘œ)
  • locality:𝑃(1)

Chinmay Nirkhe

  • Approx. QLDPC codes from spacetime Hamiltonians

To do better, we probably need to go past stabilizer codes!

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SLIDE 11

Chinmay Nirkhe

  • Approx. QLDPC codes from spacetime Hamiltonians

Going past stabilizer codes

Let 𝐼1, 𝐼2, … , 𝐼𝑛 be a set of 𝑑-local projectors acting on π‘œ qubits. Define the code-space π’Ÿ as the mutual eigenspace: π’Ÿ = πœ’ ∈ β„‚2 βŠ—π‘œ πœ’ 𝐼𝑗 πœ’ = 0 βˆ€ 𝐼𝑗

not necessarily commuting

𝐼 = 𝐼1 + β‹― + 𝐼𝑛 is 𝑑-QLDPC if additionally each qubit participates in at most 𝑑 terms 𝐼𝑗.

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SLIDE 12

Chinmay Nirkhe

  • Approx. QLDPC codes from spacetime Hamiltonians

First attempt [N-Vazirani-Yuen18]

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.

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SLIDE 13

Chinmay Nirkhe

  • Approx. QLDPC codes from spacetime Hamiltonians

First attempt [N-Vazirani-Yuen18]

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) [Kitaev99]

𝐢 𝐷

|πœ”0⟩

𝐡

|πœ”1⟩ |πœ”2⟩ |πœ”3⟩ |𝜊⟩ |0⟩

πœ”0 = 𝜊 0 πœ”1 = 𝐡 πœ”0 πœ”2 = 𝐢|πœ”1⟩ πœ”3 = 𝐷|πœ”2⟩

Together, {|πœ”π‘’βŸ©} are a β€œproof” that the circuit was executed correctly. But, ΰ·© Ξ¨ = πœ”0 πœ”1 … |πœ”π‘ˆβŸ© is not locally-checkable. Instead, the following ”clock” state* is: Ξ¨ = 1 π‘ˆ + 1 ෍

𝑒=0 π‘ˆ

𝑒 |πœ”π‘’βŸ©

*Quantum analog of Cook71-Levin73 Tableau.

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SLIDE 14

Chinmay Nirkhe

  • Approx. QLDPC codes from spacetime Hamiltonians

First attempt [N-Vazirani-Yuen18]

Create a Hamiltonian whose ground-space is almost exactly that of a CSS code but is locally checkable. Let 𝐷 = π·π‘ˆπ·π‘ˆβˆ’1 … 𝐷1 be a circuit with gates {𝐷𝑗} and let πœ”0 = |𝜊⟩|0βŸ©βŠ—π‘œβˆ’π‘™ be an initial state for |𝜊⟩ ∈ β„‚2 βŠ—π‘™. There is a local Hamiltonian with ground space of: 𝒣 = ቐ ቑ Ψ𝜊 = 1 π‘ˆ + 1 ෍

𝑒=0 π‘ˆ

unary 𝑒 βŠ— πœ”π‘’ ∢ πœ”π‘’ = 𝐷𝑒 πœ”π‘’βˆ’1 , πœ”0 = |𝜊⟩|0βŸ©βŠ—(π‘œβˆ’π‘™) .

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SLIDE 15

Chinmay Nirkhe

  • Approx. QLDPC codes from spacetime Hamiltonians

First attempt [N-Vazirani-Yuen18]

Create a Hamiltonian whose ground-space is almost exactly that of a CSS code but is locally checkable. Let π‘Š be the encoding circuit for a good CSS code. Choose 𝐿 = 𝑃 π‘ˆπ‘Šπœ€βˆ’2 .

π‘Š 𝕁

𝐿 long

𝐷

= Construct the clock Hamiltonian for this β€œpadded” circuit 𝐷.

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SLIDE 16

Chinmay Nirkhe

  • Approx. QLDPC codes from spacetime Hamiltonians

First attempt [N-Vazirani-Yuen18]

The groundspace of 𝐼 is β‰ˆ the groundspace of a CSS code tensored with junk. 𝒣𝐷 = ቐ ቑ 1 π‘ˆπ· + 1 ෍

𝑒=0 π‘ˆ

𝑒 πœ”π‘’ : πœ”π‘’ = π·π‘’π·π‘’βˆ’1 … 𝐷1 πœ”0 , πœ”0 = |𝜊⟩|0βŸ©βŠ—(π‘œβˆ’π‘™) But for 𝑒 β‰₯ π‘ˆπ‘Š, πœ”π‘’ = π‘Š πœ”0 . Thus, 1 βˆ’ 𝑃(πœ€2) fraction of πœ”π‘’ = π‘Š πœ”0 . 𝒣𝐷 β‰ˆ 1 π‘ˆπ· + 1 ෍

𝑒=0 π‘ˆ

𝑒 βŠ— ΰ΅› ࡟ π‘Š πœ”0 ∢ πœ”0 = 𝜊 0 βŠ—(π‘œβˆ’π‘™) . Plus, 𝒣𝐷 is the ground-space of a 5-local Hamiltonian! However, some qubits participate in many terms 𝐼𝑒.

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SLIDE 17

Chinmay Nirkhe

  • Approx. QLDPC codes from spacetime Hamiltonians

First attempt [N-Vazirani-Yuen18]

𝐼𝑒 checks that the slice 𝑒 |πœ”π‘’βŸ© and the slice 𝑒 + 1 πœ”π‘’+1 satisfy πœ”π‘’+1 = 𝑉𝑒|πœ”π‘’βŸ© However, some qubits participate in many terms 𝐼𝑒.

𝑒th 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.

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SLIDE 18

Chinmay Nirkhe

  • Approx. QLDPC codes from spacetime Hamiltonians

Localizing the circuit via bitonic sorting circuits

Minimize connectivity of the qubits in the circuit. Theorem [Batcher65]: There is a circuit of depth log2 π‘œ with log π‘œ connectivity sorting π‘œ elements. Can stretch circuit by log2 π‘œ mult. depth and reduce connectivity to π‘œ. Can be used anywhere to simplify circuit connectivity in any situation.

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Chinmay Nirkhe

  • Approx. QLDPC codes from spacetime Hamiltonians

Long clocks and brittle Hamiltonians

1 2 1 2 3 4

For Feynman-Kitaev clock Hamiltonian each layer of the circuit needs exactly 1 gate. This yields long clocks and brittle Hamiltonians. Brittle Hamiltonian: Small spectral gap. Not satisfying any 1equation of πœ”π‘’+1 = 𝑉𝑒|πœ”π‘’βŸ© has energy 𝑃(1/|𝐷|) with |𝐷| = the number of gates in 𝐷.

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Chinmay Nirkhe

  • Approx. QLDPC codes from spacetime Hamiltonians

Long clocks and brittle Hamiltonians

This yields long clocks and brittle Hamiltonians. There are more than |𝐷| partial computations of a circuit!

A B C D A C B C B D = 2 2 1 |1⟩

Build Hamiltonian with ground-state of uniform superposition

  • verall partial

computations 𝜐: ෍ 𝜐 |πœ”πœβŸ© Space-time Hamiltonian [Breuckmann-Terhal14]

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SLIDE 21

Chinmay Nirkhe

  • Approx. QLDPC codes from spacetime Hamiltonians

Long clocks and brittle Hamiltonians

This yields long clocks and brittle Hamiltonians. There are more than |𝐷| partial computations of a circuit!

A B C D A C B C B D = 2 2 1 |1⟩

Build Hamiltonian with ground-state of uniform superposition

  • verall partial

computations 𝜐: ෍ 𝜐 |πœ”πœβŸ© Space-time Hamiltonian [Breuckmann-Terhal14] Theorem: This yields a Hamiltonian for whom the spectral gap scales ΰ·¨ 𝑃

1 π‘œ3.09depth 𝐷 2

Instead of 𝑃

1 𝐷

as in standard Feynman-Kitaev clock Hamiltonian

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SLIDE 22

Chinmay Nirkhe

  • Approx. QLDPC codes from spacetime Hamiltonians

Long clocks and brittle Hamiltonians

This yields long clocks and brittle Hamiltonians. There are more than |𝐷| partial computations of a circuit!

A B C D A C B C B D = 2 2 1 |1⟩

Build Hamiltonian with ground-state of uniform superposition

  • verall partial

computations 𝜐: ෍ 𝜐 |πœ”πœβŸ© Space-time Hamiltonian [Breuckmann-Terhal14] Theorem: This yields a Hamiltonian for whom the spectral gap scales ΰ·¨ 𝑃

1 π‘œ3.09depth 𝐷 2

Instead of 𝑃

1 𝐷

as in standard Feynman-Kitaev clock Hamiltonian Bonus: Get 𝑃(polylog π‘œ) spatial locality

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SLIDE 23

Chinmay Nirkhe

  • Approx. QLDPC codes from spacetime Hamiltonians

Approximate decoding

Ψ = ෍

𝜐

𝜐 |πœ”π‘’βŸ©

  • ver all valid partial

computations

β‰ˆ ෍

𝜐

𝜐 βŠ— |πœ”π‘”π‘—π‘œπ‘π‘šβŸ©

due to padding with identity gates

β„° Ξ¨ β‰ˆ β„°1 ෍

𝜐

𝜐 βŠ— β„°2(|πœ”π‘”π‘—π‘œπ‘π‘šβŸ©)

Approximate decoding:

  • 1. Trace out clock

registers

  • 2. Apply underlying

code decoding procedure

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Chinmay Nirkhe

  • Approx. QLDPC codes from spacetime Hamiltonians

Construction recap

A code with linear rate and distance and 𝑃(log3 π‘œ) depth encoding circuit [Brown-Fawzi13] Uniformize the connectivity of the circuit using bitonic sorting circuits Build spacetime Hamiltonian of resulting code [Breuckmann-Terhal14] Pad the circuit with identity gates

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SLIDE 25

Chinmay Nirkhe

  • Approx. QLDPC codes from spacetime Hamiltonians

Spectral gap analysis

Def: minimum non-zero eigenvalue of Hamiltonian 𝐼 Map the Hamiltonian to a Markov chain over the space of valid partial computations

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SLIDE 26

Chinmay Nirkhe

  • Approx. QLDPC codes from spacetime Hamiltonians

Spectral gap analysis

Spectral gap of the code is based on the mixing time of valid configurations

  • f a bitonic block

True of all constructions built from bitonic sorting circuits

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SLIDE 27

Chinmay Nirkhe

  • Approx. QLDPC codes from spacetime Hamiltonians

Spectral gap analysis

Spectral gap of the code is based on the mixing time of valid configurations

  • f a bitonic block

We noticed that bitonic blocks look similar to a structure called dyadic tilings studied in [Cannon-Levin-Stauffer17] Dyadic tilings are ways of covering the unit square by 2𝑒 rectangles with corner coordinates at multiples of 2βˆ’π‘’

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SLIDE 28

Chinmay Nirkhe

  • Approx. QLDPC codes from spacetime Hamiltonians

Spectral gap analysis

Spectral gap of the code is based on the mixing time of valid configurations

  • f a bitonic block
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SLIDE 29

Chinmay Nirkhe

  • Approx. QLDPC codes from spacetime Hamiltonians

Spectral gap analysis

Spectral gap of the code is based on the mixing time of valid configurations

  • f a bitonic block
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SLIDE 30

Chinmay Nirkhe

  • Approx. QLDPC codes from spacetime Hamiltonians

Spectral gap analysis

Spectral gap of the code is based on the mixing time of valid configurations

  • f a bitonic block
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SLIDE 31

Chinmay Nirkhe

  • Approx. QLDPC codes from spacetime Hamiltonians

Spectral gap analysis

Spectral gap of the code is based on the mixing time of valid configurations

  • f a bitonic block
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SLIDE 32

Chinmay Nirkhe

  • Approx. QLDPC codes from spacetime Hamiltonians

Spectral gap analysis

Spectral gap of the code is based on the mixing time of valid configurations

  • f a bitonic block

Theorem: The spectral gap of this Hamiltonian is ΰ·© Ξ© (π‘œβˆ’3.09).

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SLIDE 33

Chinmay Nirkhe

  • Approx. QLDPC codes from spacetime Hamiltonians

Summary of results

We constructed a new type of code based on spacetime Hamiltonians. It has the following properties:

  • rate: Ξ©(

1 polylog π‘œ)

  • distance:Ξ©(

π‘œ polylog π‘œ)

  • spatial-locality: Ξ© polylog π‘œ
  • spectral-gap: Ξ©(π‘œβˆ’3.09)

Along the way, we also learned about

  • localizing large stabilizers using

circuit-to-Hamiltonian constructions

  • uniformizing circuits with bitonic

sorting networks

  • analysis of uniform circuits via

Markov chain techniques

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SLIDE 34

Chinmay Nirkhe

  • Approx. QLDPC codes from spacetime Hamiltonians

What does this teach us?

First, this isn’t the β€˜β€™perfect’’ error-correcting code or is realistic Relaxing the requirements of stabilizer codes is helpful

  • Code-space as the ground-space of a sum of non-commuting projectors
  • Approximate error-correction

There are connections between computation and error- correction that we don’t fully understand!

Thanks!