Lecture Notes on Quantum Algorithms

Lecture Notes onQuantum AlgorithmsAndrew M. ChildsDepartment of Computer Science,Institute for Advanced Computer Studies, andJoint Center for Quantum Information and Computer ScienceUniversity of Maryland29 April 2021iiContentsPrefacevii1 Quantum data .. Quantum circuits .. Universal gate sets .. Reversible computation .. Uniformity .. Quantum complexity .. Fault tolerance ..3I Quantum circuits52 Efficient universality of Quantum Subadditivity of errors .. The group commutator and a net around the identity .. Proof of the Solovay-Kitaev Theorem .. Proof of Lemma ..93 Quantum circuit synthesis over Clifford+ Converting to Matsumoto-Amano normal form .. Uniqueness of Matsumoto-Amano normal form .. Algebraic characterization of Clifford+Tunitaries .. From exact to approximate synthesis ..14II Quantum Algorithms for algebraic problems154 The abelian Quantum Fourier transform and phase Quantum Fourier transform.

for an arbitrary superposition over the group. We assume that there is some canonical way of e ciently representing group elements using bit strings; it is usually unnecessary to make this representation explicit. If a quantum computer stores the state j iand the state j˚i, its overall state is given by the tensor product of those two states.

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