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Mixed States and Noise

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Density matrix recap

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Density matrices of multiple systems

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Traces and partial traces

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Traces and partial traces

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Density matrices

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Traces and partial traces

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Distinguishability of density matrices

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Distinguishability of density matrices

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Density matrices

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Modeling noise

  • A cosmic ray comes from outer space and perturbs some of the qubits in the middle of a computation. How to describe the state of �your quantum computer?

 

 

 

No noise:

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Modeling noise

  • A cosmic ray comes from outer space and perturbs some of the qubits in the middle of a computation. How to describe the state of �your quantum computer?

 

 

 

With noise:

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Principle of Deferred Measurement

  • A circuit with intermediate measurements can always be converted into a circuit where measurements happen at the end (i.e. they are ”deferred”)
  • For every intermediate measurement, replace with a CNOT into fresh ancilla qubit.

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And now for something different!

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The Measurement Problem

  • Physicists and philosophers have long puzzled over why quantum mechanics has two different ways a state can change: unitaries and measurements.

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  • What is so special about the act of measurement?

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  • How exactly does a state “collapse”?

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  • Who is doing the measurement?

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The Measurement Problem

  • Lots of spilled ink and theories on the Measurement Problem, including:

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    • Copenhagen Interpretation: “Shut up and calculate!”

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    • De Broglie-Bohm Interpretation

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    • Hugh Everett’s Many-Worlds Interpretation

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    • Many others…

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The Measurement Problem

  • The Many-Worlds Interpretation

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    • There is only unitary evolution. The universe is in a pure state.

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    • There is no difference between the observer and the observed – we all live in the same pure state.

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    • “Measurement” is simply a unitary interaction �between the observer and the system.

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    • Each branch of the superposition corresponds to a �different outcome – a different “world”.

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The Measurement Problem

Schrodinger’s Cat thought experiment, using the Many-Worlds Interpretation:

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Three objects:

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…back to our regularly scheduled quantum

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Quantum operations

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Quantum operations

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Quantum operations

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Quantum operations

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Quantum operations

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Noise operations/channels

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Representations of quantum operations

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Next time

Dealing with errors in a quantum computation