Memory Hierarchies for Quantum Data

1
Memory Hierarchies for Quantum Data

• Dean Copsey, Mark Oskin, Frederic T. Chong, Isaac
  Chaung and Khaled Abdel-Ghaffar
• Presented by Greg Gerou

2
Introduction

• Environmental noise is a big problem qubits are
  easily influenced by factors within and without
  the computer.
• Threshold theorem As long as the probability n
  of error of each operations on a quantum computer
  is less than some constant (estimated to be as
  high as 10-4), scalable quantum computers can be
  built using faulty components.

3
Introduction (contd)

• Error correction codes have been developed to
  establish different levels of reliability, but
  there are overhead trade-offs.
• The goal of this paper is to reduce the overhead
  of error correction for the memory system.

4
Basic Quantum Operations
(
)
( )
5
Controlled Entanglement
This figure demonstrates the entanglement of two
bits. The value of the two qubits are linked,
ensuring that the bits will be either 11 or 00
(the probability amplitudes for 01 and 10 are
zero).
The interaction between the two bits determines
their probability amplitudes. Similarly, the
outside environment has a significant impact on
the probability amplitudes of our qubits.
6
Uncontrolled Entanglement

• Electrons emit and absorb photons, changing their
  orbitals.
• Magnetic spin states of nuclei can be flipped by
  external magnetic fields.
• Due to entanglement with the environment, its
  impossible to isolate a system to the point where
  it is completely stable.
• This introduction of error due to uncontrolled
  entanglement is termed decoherence.

7
Quantum Error Correction

• A logical qubit can be encoded using a number of
  physical qubits.
• Encoding size constraints are driven by the two
  types of error correction
• Amplitude correction
• Phase correction
• Three bit error correction (Shor code)

8
Quantum Error Correction

• Given that
• We must correct for error in both phase and
  amplitude.
• Using Shor code, 3 bits are required to perform
  either phase or amplitude correction.
• Once we perform an error correction, our source
  bits are put into a different state.
• Shors code requires that one logical qubit be
  encoded into 3 bits for error correction, and
  those three bits each need to be encoded into
  three bits for amplitude correction.

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Quantum Error Correction
Phase corrected qubits
Logical qubit (corrected for both phase and
amplitude)
Uncorrected qubit vector
10
Quantum Error Correction

• Shors code is termed a 9,1,3 code
• Nine physical qubits
• One logical qubit
• Three is the Hamming distance
• A code with a Hamming distance of d is able to
  correct (d-1)/2 errors. In this case, one error
  can be corrected.

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Other Encodings

• Stabalizer code 5,1,3 (densest known way to
  encode a single qubit)
• 8,3,3 (densest known three qubit code)
• Steanes 7,1,3. This code is nice
• Operators can be applied to the logical bits by
  applying simple operators on the physical bits.
  For example, to perform a NOT on a logical bit,
  it is only necessary to perform a NOT on each of
  the physical bits.

12
Error Calculations

• As long as the probability, p, of an error is
  below a certain threshold, c (10-14 in the case
  of Steanes code), any number of operations can
  be performed with the probability of error
• cp2

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Concatenation

• If a single logical qubit is encoded by seven
  (Steanes code) physical qubits, what happens to
  the error if we encode each of those seven?
• c(cp2)2 ltlt cp2

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Concatenation Example

• 7,1,3 concatenated once

This logical qubit
is encoded by these seven qubits
each of which is encoded by its own seven
physical qubits.
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Concatenation

• The circuit size and time complexity is growing
  exponentially! Say we concatenate k times
• Time tk
• Circuit size dk
• However, error is reduced significantly also

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Concatenation
Overheads for different recursion levels of
7,1,3
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Teleportation

• Definition The re-creation of a quantum state
  at a destination using some classical bits that
  must be communicated along conventional wires or
  other mediums.
• Teleportation is key in converting between
  different types of encodings, and in transferring
  memory.

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Memory Hierarchies

• Idea Use different encodings at different levels
  of memory
• Large encodings are good for CPU memory
• Disadvantages
• Take a lot of space (many physical qubits)
• Advantages
• Better error correction
• Smaller encodings are good for storage
• Disadvantages
• Worse error correction
• Advantages
• Much more dense (fewer physical qubits)

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Memory Hierarchy Encoding levels
Overhead per logical qubit
Encoding Physical qubits
343,1,15 343
245,1,15 245
392,3,15 131
Note also that teleportation is relatively slow.
This implies that there is a time penalty when
data is moved from one level of memory to another.
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Memory Hierarchy

• We can take advantage of temporal and spatial
  locality. For instance, take the following
  nine-bit Quantum Fourier Transfer (QFT)

Cost 9 logical qubits 343 physical qubits per
bit 3,087 physical qubits
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Memory Hierarchy

• Now lets reorder the operations and use a cache

Cost (6 logical 343 physical) (3 logical
131 physical) 2,451 physical
22
Memory Hierarchy

• 2,451 physical qubits may not seem like a huge
  advantage over 3,087, but another way to look at
  it is the processor will contain 60 fewer
  physical bits.
• Take also into account that the data in the cache
  will not be operated on nearly as much as the
  data in the CPU, implying much less decoherence
  (and so smaller error correction requirements).

23
Future Work

• There also exist non-concatenated codes that
  offer improved density and possibly improved
  performance.
• An dependency on what codes are used for each of
  the memory hierarchies is the physical properties
  of the quantum system
• How much error is introduced by the environment?
• How fast can it operate?