Quantum Circuit Placement: Optimizing Qubit-to-qubit Interactions through Mapping Quantum Circuits into a Physical Experiment D. Maslov (spkr) – IQC/UWaterloo, Canada S. M. Falconer – UofVictoria, Canada M. Mosca – IQC/UWaterloo, Canada

Outline - Why quantum computing? - Background - Circuit placement technique - Results page 1/10

Why quantum computing? page 2/10 Algorithms: Abelian stabiliser, Bernstein-Vazirani parity problem, Deutsch-Jozsa, differential equations, discrete logarithm, Eigenvalues, factoring, group theory, Grover (database search), hidden subgroup problem, Jones polynomials, Median, NAND tree evaluation, numerical integration, pattern recognition, physical simulation, Poincare recurrences, polynomial shift equivalence, random walks, random number generation, Simon, etc. AlgorithmsImplementation

Why quantum computing? page 2/10 Algorithms: Abelian stabiliser, Bernstein-Vazirani parity problem, Deutsch-Jozsa, differential equations, discrete logarithm, Eigenvalues, factoring, group theory, Grover (database search), hidden subgroup problem, Jones polynomials, Median, NAND tree evaluation, numerical integration, pattern recognition, physical simulation, Poincare recurrences, polynomial shift equivalence, random walks, random number generation, Simon, etc. Complexity separation AlgorithmsImplementation

Why quantum computing? page 2/10 Algorithms: Abelian stabiliser, Bernstein-Vazirani parity problem, Deutsch-Jozsa, differential equations, discrete logarithm, Eigenvalues, factoring, group theory, Grover (database search), hidden subgroup problem, Jones polynomials, Median, NAND tree evaluation, numerical integration, pattern recognition, physical simulation, Poincare recurrences, polynomial shift equivalence, random walks, random number generation, Simon, etc. Complexity separation Famous AlgorithmsImplementation

Why quantum computing? page 2/10 Algorithms: Abelian stabiliser, Bernstein-Vazirani parity problem, Deutsch-Jozsa, differential equations, discrete logarithm, Eigenvalues, factoring, group theory, Grover (database search), hidden subgroup problem, Jones polynomials, Median, NAND tree evaluation, numerical integration, pattern recognition, physical simulation, Poincare recurrences, polynomial shift equivalence, random walks, random number generation, Simon, etc. Complexity separation Famous Most Practical AlgorithmsImplementation

Why quantum computing? page 2/10 Algorithms: Abelian stabiliser, Bernstein-Vazirani parity problem, Deutsch-Jozsa, differential equations, discrete logarithm, Eigenvalues, factoring, group theory, Grover (database search), hidden subgroup problem, Jones polynomials, Median, NAND tree evaluation, numerical integration, pattern recognition, physical simulation, Poincare recurrences, polynomial shift equivalence, random walks, random number generation, Simon, etc. Complexity separation Famous Most Practical Commercially Available AlgorithmsImplementation

Why quantum computing? page 3/10 Implementation: liquid Nuclear Magnetic Resonance, solid NMR, ion traps, neutral atoms, cavity QED, optic technologies, solid state, superconducting (Josephson junctions), etc. AlgorithmsImplementation

Why quantum computing? page 3/10 Implementation: liquid Nuclear Magnetic Resonance, solid NMR, ion traps, neutral atoms, cavity QED, optic technologies, solid state, superconducting (Josephson junctions), etc. Most developed AlgorithmsImplementation

Why quantum computing? page 3/10 Implementation: liquid Nuclear Magnetic Resonance, solid NMR, ion traps, neutral atoms, cavity QED, optic technologies, solid state, superconducting (Josephson junctions), etc. Most developed Likely most promising AlgorithmsImplementation

Why quantum computing? page 4/10 AlgorithmsImplementation Quantum CAD tools Quantum CAD tools: circuit optimization, error correction, synthesis, testing, verification. In this work: placement.

Background page 5/10 Algorithms are circuits. time

Background page 6/10 Histidine Hydrogen Nitrogen Carbon Implementation: Liquid NMR Communication via chemical bonds.

Circuit placement page 7/10 Problem: given a circuit and a quantum mechanical system, assign qubits as to optimize the runtime. Theorem: the above problem is NP-Complete. Additional complication: it might be possible to place subcircuits nicely, but the total runtime is high. Solution: Place subcircuits Permute qubit assignments

Circuit placement page 8/10 Place subcircuits Permute qubit assignments

Circuit placement page 8/10 Place subcircuits Permute qubit assignments Circuit is a graph: nodes are logic qubits, edges are two-qubit gates. Physical implementation is a graph: nodes are physical qubits (e.g., nuclei), edges are “fast” connections (e.g., chemical bonds). Solution: place subcircuits using graph monomorphism techniques.

Circuit placement page 8/10 Permute qubit assignments

Circuit placement page 9/10 Permute qubit assignments Problem: given two qubit-to-nuclei assignments, permute one into the other. Trans-crotonic acid

Circuit placement page 9/10 Permute qubit assignments Cut the graph into two maximally balanced connected components. BLACK WHITE Color vertices black and white according to which subgraph we want to permute their values to.

Circuit placement page 9/10 Permute qubit assignments Cut the graph into two maximally balanced connected components. BLACK WHITE Color vertices black and white according to which subgraph we want to permute their values to.

Circuit placement page 9/10 Permute qubit assignments

Circuit placement page 9/10 Permute qubit assignments White is air, black is water. Step 1: SWAP(4,5) SWAP(6,7)

Circuit placement page 9/10 Permute qubit assignments White bubbles rise.Step 1: SWAP(4,5) SWAP(6,7) Step 2: SWAP(3,4)

Circuit placement page 9/10 Permute qubit assignments White bubbles rise.Step 1: SWAP(4,5) SWAP(6,7) Step 2: SWAP(3,4) Step 3: SWAP(2,3) SWAP(4,6)

Circuit placement page 9/10 Permute qubit assignments White bubbles rise.Step 1: SWAP(4,5) SWAP(6,7) Step 2: SWAP(3,4) Step 3: SWAP(2,3) SWAP(4,6) Step 4: SWAP(3,4)

Circuit placement page 9/10 Permute qubit assignments The problem falls into two. We proved that with some natural restrictions the solution has linear depth.

Results page 10/10 error correct. encoding93acetyl chloride sec error correction255trans-crotonic acid sec pseudo cat state prep.5410histidine sec Circ.: name, gates, qubits Impl.: name, qubits Runtime Placement is identical to that by experimentalists. # of subcircuitsSwapping timeComputation timeTotal runtime 10 sec.4137 sec sec.0298 sec.3622 sec sec.0201 sec.7048 sec 6-qubit quantum Fourier transform/trans-crotonic acid.

END Thank you for your attention!