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Preparing Ground States of Quantum Many-Body Systems on a Quantum Computer David Poulin Département de Physique Université de Sherbrooke Canadian Mathematical Society Ottawa, December 2008 David Poulin (Sherbrooke) Preparing Ground States CMS’08 1 / 19 Motivation Outline 1 Motivation 2 Existing methods 3 The obvious method 4 An algorithm to prepare ground states in time 5 Applications David Poulin (Sherbrooke) Preparing Ground States √ 2n CMS’08 2 / 19 Motivation Quantum simulation Many-body problem Hilbert space: H = (C2 )⊗n . P Hamiltonian: H = hi,ji hij where Hermitian hij is the identity operator except on the ith and jth tensor factor. Dynamics: ψ ∈ H, d dt ψ = −iHψ. Solution ψ(t) = e−iHt ψ(0). Ground state: eigenstate associated to lowest eigenvalue of H. Efficient simulation with a quantum computer. David Poulin (Sherbrooke) Preparing Ground States CMS’08 3 / 19 Motivation Quantum simulation Many-body problem Hilbert space: H = (C2 )⊗n . P Hamiltonian: H = hi,ji hij where Hermitian hij is the identity operator except on the ith and jth tensor factor. Dynamics: ψ ∈ H, d dt ψ = −iHψ. Solution ψ(t) = e−iHt ψ(0). Ground state: eigenstate associated to lowest eigenvalue of H. Efficient simulation with a quantum computer. David Poulin (Sherbrooke) Preparing Ground States CMS’08 3 / 19 Motivation Quantum simulation Many-body problem Hilbert space: H = (C2 )⊗n . P Hamiltonian: H = hi,ji hij where Hermitian hij is the identity operator except on the ith and jth tensor factor. Dynamics: ψ ∈ H, d dt ψ = −iHψ. Solution ψ(t) = e−iHt ψ(0). Ground state: eigenstate associated to lowest eigenvalue of H. Efficient simulation with a quantum computer. David Poulin (Sherbrooke) Preparing Ground States CMS’08 3 / 19 Motivation Quantum simulation Many-body problem Hilbert space: H = (C2 )⊗n . P Hamiltonian: H = hi,ji hij where Hermitian hij is the identity operator except on the ith and jth tensor factor. Dynamics: ψ ∈ H, d dt ψ = −iHψ. Solution ψ(t) = e−iHt ψ(0). Ground state: eigenstate associated to lowest eigenvalue of H. Efficient simulation with a quantum computer. David Poulin (Sherbrooke) Preparing Ground States CMS’08 3 / 19 Motivation Quantum simulation Many-body problem Hilbert space: H = (C2 )⊗n . P Hamiltonian: H = hi,ji hij where Hermitian hij is the identity operator except on the ith and jth tensor factor. Dynamics: ψ ∈ H, d dt ψ = −iHψ. Solution ψ(t) = e−iHt ψ(0). Ground state: eigenstate associated to lowest eigenvalue of H. Efficient simulation with a quantum computer. David Poulin (Sherbrooke) Preparing Ground States CMS’08 3 / 19 Motivation Quantum simulation Many-body problem Hilbert space: H = (C2 )⊗n . P Hamiltonian: H = hi,ji hij where Hermitian hij is the identity operator except on the ith and jth tensor factor. Dynamics: ψ ∈ H, d dt ψ = −iHψ. Solution ψ(t) = e−iHt ψ(0). Ground state: eigenstate associated to lowest eigenvalue of H. Efficient simulation with a quantum computer. A e-iHt = U # of gates = poly(n,t) David Poulin (Sherbrooke) Preparing Ground States CMS’08 3 / 19 Motivation A notoriously hard problem This ’integration tool’ is not of much use if we cannot specify the initial conditions; what is ψ(0)? Physically relevant are the low energy states of H. Preparing them is a difficult problem: Question Is the lowest eigenvalue of H less than α or greater than β where β − α > 1/poly (n). Classical case, i.e. [hij , hkl ] = 0, is NP-complete (Barahona). Quantum case is QMA-complete (Kitaev). David Poulin (Sherbrooke) Preparing Ground States CMS’08 4 / 19 Motivation A notoriously hard problem This ’integration tool’ is not of much use if we cannot specify the initial conditions; what is ψ(0)? Physically relevant are the low energy states of H. Preparing them is a difficult problem: Question Is the lowest eigenvalue of H less than α or greater than β where β − α > 1/poly (n). Classical case, i.e. [hij , hkl ] = 0, is NP-complete (Barahona). Quantum case is QMA-complete (Kitaev). David Poulin (Sherbrooke) Preparing Ground States CMS’08 4 / 19 Motivation A notoriously hard problem This ’integration tool’ is not of much use if we cannot specify the initial conditions; what is ψ(0)? Physically relevant are the low energy states of H. Preparing them is a difficult problem: Question Is the lowest eigenvalue of H less than α or greater than β where β − α > 1/poly (n). Classical case, i.e. [hij , hkl ] = 0, is NP-complete (Barahona). Quantum case is QMA-complete (Kitaev). David Poulin (Sherbrooke) Preparing Ground States CMS’08 4 / 19 Motivation A notoriously hard problem This ’integration tool’ is not of much use if we cannot specify the initial conditions; what is ψ(0)? Physically relevant are the low energy states of H. Preparing them is a difficult problem: Question Is the lowest eigenvalue of H less than α or greater than β where β − α > 1/poly (n). Classical case, i.e. [hij , hkl ] = 0, is NP-complete (Barahona). Quantum case is QMA-complete (Kitaev). David Poulin (Sherbrooke) Preparing Ground States CMS’08 4 / 19 Motivation A notoriously hard problem This ’integration tool’ is not of much use if we cannot specify the initial conditions; what is ψ(0)? Physically relevant are the low energy states of H. Preparing them is a difficult problem: Question Is the lowest eigenvalue of H less than α or greater than β where β − α > 1/poly (n). Classical case, i.e. [hij , hkl ] = 0, is NP-complete (Barahona). Quantum case is QMA-complete (Kitaev). David Poulin (Sherbrooke) Preparing Ground States CMS’08 4 / 19 Existing methods Outline 1 Motivation 2 Existing methods 3 The obvious method 4 An algorithm to prepare ground states in time 5 Applications David Poulin (Sherbrooke) Preparing Ground States √ 2n CMS’08 5 / 19 Existing methods Simulated Annealing Used for all sort of (classical) combinatorial optimization problems. Imitates an initially hot metal (random state) that is slowly cooled down. If cooling is too fast, the system can become trapped in a local minimum. Running time depends on the energy landscape. In the worst case, the running time is proportional to the number of states N = 2n . √ A quantum simulated annealing algorithm offers a · speed-up over all classical annealing processes. (Somma et al.) These techniques are only suited for classical problems. David Poulin (Sherbrooke) Preparing Ground States CMS’08 6 / 19 Existing methods Simulated Annealing Used for all sort of (classical) combinatorial optimization problems. Imitates an initially hot metal (random state) that is slowly cooled down. If cooling is too fast, the system can become trapped in a local minimum. Running time depends on the energy landscape. In the worst case, the running time is proportional to the number of states N = 2n . √ A quantum simulated annealing algorithm offers a · speed-up over all classical annealing processes. (Somma et al.) These techniques are only suited for classical problems. David Poulin (Sherbrooke) Preparing Ground States CMS’08 6 / 19 Existing methods Simulated Annealing Used for all sort of (classical) combinatorial optimization problems. Imitates an initially hot metal (random state) that is slowly cooled down. If cooling is too fast, the system can become trapped in a local minimum. Running time depends on the energy landscape. In the worst case, the running time is proportional to the number of states N = 2n . √ A quantum simulated annealing algorithm offers a · speed-up over all classical annealing processes. (Somma et al.) These techniques are only suited for classical problems. David Poulin (Sherbrooke) Preparing Ground States CMS’08 6 / 19 Existing methods Simulated Annealing Used for all sort of (classical) combinatorial optimization problems. Imitates an initially hot metal (random state) that is slowly cooled down. If cooling is too fast, the system can become trapped in a local minimum. Running time depends on the energy landscape. In the worst case, the running time is proportional to the number of states N = 2n . √ A quantum simulated annealing algorithm offers a · speed-up over all classical annealing processes. (Somma et al.) These techniques are only suited for classical problems. David Poulin (Sherbrooke) Preparing Ground States CMS’08 6 / 19 Existing methods Simulated Annealing Used for all sort of (classical) combinatorial optimization problems. Imitates an initially hot metal (random state) that is slowly cooled down. If cooling is too fast, the system can become trapped in a local minimum. Running time depends on the energy landscape. In the worst case, the running time is proportional to the number of states N = 2n . √ A quantum simulated annealing algorithm offers a · speed-up over all classical annealing processes. (Somma et al.) These techniques are only suited for classical problems. David Poulin (Sherbrooke) Preparing Ground States CMS’08 6 / 19 Existing methods Simulated Annealing Used for all sort of (classical) combinatorial optimization problems. Imitates an initially hot metal (random state) that is slowly cooled down. If cooling is too fast, the system can become trapped in a local minimum. Running time depends on the energy landscape. In the worst case, the running time is proportional to the number of states N = 2n . √ A quantum simulated annealing algorithm offers a · speed-up over all classical annealing processes. (Somma et al.) These techniques are only suited for classical problems. David Poulin (Sherbrooke) Preparing Ground States CMS’08 6 / 19 Existing methods Adiabatic quantum computing Theorem (Adiabatic evolution) A quantum system will remain in its instantaneous ground state if the Hamiltonian that governs its evolution varies slowly enough. (T − t) t Hhard + Heasy T T At t = 0, prepare the system in the ground state of Heasy . H(t) = Success if T ≥ (mint gap{H(t)})−1 where gap{H} = λ1 − λ0 . For classical problems, in the worst case the gap is 1/2n . (van Dam & Vazirani) Not studied for the quantum case. David Poulin (Sherbrooke) Preparing Ground States CMS’08 7 / 19 Existing methods Adiabatic quantum computing Theorem (Adiabatic evolution) A quantum system will remain in its instantaneous ground state if the Hamiltonian that governs its evolution varies slowly enough. (T − t) t Hhard + Heasy T T At t = 0, prepare the system in the ground state of Heasy . H(t) = Success if T ≥ (mint gap{H(t)})−1 where gap{H} = λ1 − λ0 . For classical problems, in the worst case the gap is 1/2n . (van Dam & Vazirani) Not studied for the quantum case. David Poulin (Sherbrooke) Preparing Ground States CMS’08 7 / 19 Existing methods Adiabatic quantum computing Theorem (Adiabatic evolution) A quantum system will remain in its instantaneous ground state if the Hamiltonian that governs its evolution varies slowly enough. (T − t) t Hhard + Heasy T T At t = 0, prepare the system in the ground state of Heasy . H(t) = Success if T ≥ (mint gap{H(t)})−1 where gap{H} = λ1 − λ0 . For classical problems, in the worst case the gap is 1/2n . (van Dam & Vazirani) Not studied for the quantum case. David Poulin (Sherbrooke) Preparing Ground States CMS’08 7 / 19 Existing methods Adiabatic quantum computing Theorem (Adiabatic evolution) A quantum system will remain in its instantaneous ground state if the Hamiltonian that governs its evolution varies slowly enough. (T − t) t Hhard + Heasy T T At t = 0, prepare the system in the ground state of Heasy . H(t) = Success if T ≥ (mint gap{H(t)})−1 where gap{H} = λ1 − λ0 . For classical problems, in the worst case the gap is 1/2n . (van Dam & Vazirani) Not studied for the quantum case. David Poulin (Sherbrooke) Preparing Ground States CMS’08 7 / 19 Existing methods Adiabatic quantum computing Theorem (Adiabatic evolution) A quantum system will remain in its instantaneous ground state if the Hamiltonian that governs its evolution varies slowly enough. (T − t) t Hhard + Heasy T T At t = 0, prepare the system in the ground state of Heasy . H(t) = Success if T ≥ (mint gap{H(t)})−1 where gap{H} = λ1 − λ0 . For classical problems, in the worst case the gap is 1/2n . (van Dam & Vazirani) Not studied for the quantum case. David Poulin (Sherbrooke) Preparing Ground States CMS’08 7 / 19 Existing methods Grover’s algorithm Given a projector P onto a subspace of H and a state ψ, construct the two reflections RP = I − 2P and Rψ = 1 − 2|ψihψ|. Then kP(RP Rψ )t |ψik2 ∈ O(1) when t = dkP|ψik−1 2 e. Black box interpretation Maps state |ψi to 1 kP|ψik P|ψi in time 1 kP|ψik . Choose P to be the projection on the ’solution’, i.e. onto H < α. q −1 Choose ψ at random (2-design) so that kP|ψik2 = rank2n(P) . √ Then the solution is found in t ≤ 2n iterations. The projector onto H < α is easy to construct for classical H. How to construct it for quantum systems? David Poulin (Sherbrooke) Preparing Ground States CMS’08 8 / 19 Existing methods Grover’s algorithm Given a projector P onto a subspace of H and a state ψ, construct the two reflections RP = I − 2P and Rψ = 1 − 2|ψihψ|. Then kP(RP Rψ )t |ψik2 ∈ O(1) when t = dkP|ψik−1 2 e. Black box interpretation Maps state |ψi to 1 kP|ψik P|ψi in time 1 kP|ψik . Choose P to be the projection on the ’solution’, i.e. onto H < α. q −1 Choose ψ at random (2-design) so that kP|ψik2 = rank2n(P) . √ Then the solution is found in t ≤ 2n iterations. The projector onto H < α is easy to construct for classical H. How to construct it for quantum systems? David Poulin (Sherbrooke) Preparing Ground States CMS’08 8 / 19 Existing methods Grover’s algorithm Given a projector P onto a subspace of H and a state ψ, construct the two reflections RP = I − 2P and Rψ = 1 − 2|ψihψ|. Then kP(RP Rψ )t |ψik2 ∈ O(1) when t = dkP|ψik−1 2 e. Black box interpretation Maps state |ψi to 1 kP|ψik P|ψi in time 1 kP|ψik . Choose P to be the projection on the ’solution’, i.e. onto H < α. q −1 Choose ψ at random (2-design) so that kP|ψik2 = rank2n(P) . √ Then the solution is found in t ≤ 2n iterations. The projector onto H < α is easy to construct for classical H. How to construct it for quantum systems? David Poulin (Sherbrooke) Preparing Ground States CMS’08 8 / 19 Existing methods Grover’s algorithm Given a projector P onto a subspace of H and a state ψ, construct the two reflections RP = I − 2P and Rψ = 1 − 2|ψihψ|. Then kP(RP Rψ )t |ψik2 ∈ O(1) when t = dkP|ψik−1 2 e. Black box interpretation Maps state |ψi to 1 kP|ψik P|ψi in time 1 kP|ψik . Choose P to be the projection on the ’solution’, i.e. onto H < α. q −1 Choose ψ at random (2-design) so that kP|ψik2 = rank2n(P) . √ Then the solution is found in t ≤ 2n iterations. The projector onto H < α is easy to construct for classical H. How to construct it for quantum systems? David Poulin (Sherbrooke) Preparing Ground States CMS’08 8 / 19 Existing methods Grover’s algorithm Given a projector P onto a subspace of H and a state ψ, construct the two reflections RP = I − 2P and Rψ = 1 − 2|ψihψ|. Then kP(RP Rψ )t |ψik2 ∈ O(1) when t = dkP|ψik−1 2 e. Black box interpretation Maps state |ψi to 1 kP|ψik P|ψi in time 1 kP|ψik . Choose P to be the projection on the ’solution’, i.e. onto H < α. q −1 Choose ψ at random (2-design) so that kP|ψik2 = rank2n(P) . √ Then the solution is found in t ≤ 2n iterations. The projector onto H < α is easy to construct for classical H. How to construct it for quantum systems? David Poulin (Sherbrooke) Preparing Ground States CMS’08 8 / 19 Existing methods Grover’s algorithm Given a projector P onto a subspace of H and a state ψ, construct the two reflections RP = I − 2P and Rψ = 1 − 2|ψihψ|. Then kP(RP Rψ )t |ψik2 ∈ O(1) when t = dkP|ψik−1 2 e. Black box interpretation Maps state |ψi to 1 kP|ψik P|ψi in time 1 kP|ψik . Choose P to be the projection on the ’solution’, i.e. onto H < α. q −1 Choose ψ at random (2-design) so that kP|ψik2 = rank2n(P) . √ Then the solution is found in t ≤ 2n iterations. The projector onto H < α is easy to construct for classical H. How to construct it for quantum systems? David Poulin (Sherbrooke) Preparing Ground States CMS’08 8 / 19 Existing methods Grover’s algorithm Given a projector P onto a subspace of H and a state ψ, construct the two reflections RP = I − 2P and Rψ = 1 − 2|ψihψ|. Then kP(RP Rψ )t |ψik2 ∈ O(1) when t = dkP|ψik−1 2 e. Black box interpretation Maps state |ψi to 1 kP|ψik P|ψi in time 1 kP|ψik . Choose P to be the projection on the ’solution’, i.e. onto H < α. q −1 Choose ψ at random (2-design) so that kP|ψik2 = rank2n(P) . √ Then the solution is found in t ≤ 2n iterations. The projector onto H < α is easy to construct for classical H. How to construct it for quantum systems? David Poulin (Sherbrooke) Preparing Ground States CMS’08 8 / 19 Existing methods Grover’s algorithm Given a projector P onto a subspace of H and a state ψ, construct the two reflections RP = I − 2P and Rψ = 1 − 2|ψihψ|. Then kP(RP Rψ )t |ψik2 ∈ O(1) when t = dkP|ψik−1 2 e. Black box interpretation Maps state |ψi to 1 kP|ψik P|ψi in time 1 kP|ψik . Choose P to be the projection on the ’solution’, i.e. onto H < α. q −1 Choose ψ at random (2-design) so that kP|ψik2 = rank2n(P) . √ Then the solution is found in t ≤ 2n iterations. The projector onto H < α is easy to construct for classical H. How to construct it for quantum systems? David Poulin (Sherbrooke) Preparing Ground States CMS’08 8 / 19 The obvious method Outline 1 Motivation 2 Existing methods 3 The obvious method 4 An algorithm to prepare ground states in time 5 Applications David Poulin (Sherbrooke) Preparing Ground States √ 2n CMS’08 9 / 19 The obvious method Phase estimation Ingredients An efficiently implementable unitary matrix U on n qubits. Spectral decomposition U|ai = ei2πϕa |ai. k = polylog(n) auxiliary qubits in the state |0k i. Phase estimation maps |ai ⊗ |0k i → |ai ⊗ |ϕa i in poly time where X |ϕa i = ei2πϕa j |ji. j These are ’momentum’ states so the value of ϕa can be measured via Fourier transform. Substituting U = e−iHt with t < kHk−1 provides a method to measure the energy and hence to construct a projector onto H < α for use in Grover’s algorithm. David Poulin (Sherbrooke) Preparing Ground States CMS’08 10 / 19 The obvious method Phase estimation Ingredients An efficiently implementable unitary matrix U on n qubits. Spectral decomposition U|ai = ei2πϕa |ai. k = polylog(n) auxiliary qubits in the state |0k i. Phase estimation maps |ai ⊗ |0k i → |ai ⊗ |ϕa i in poly time where X |ϕa i = ei2πϕa j |ji. j These are ’momentum’ states so the value of ϕa can be measured via Fourier transform. Substituting U = e−iHt with t < kHk−1 provides a method to measure the energy and hence to construct a projector onto H < α for use in Grover’s algorithm. David Poulin (Sherbrooke) Preparing Ground States CMS’08 10 / 19 The obvious method Phase estimation Ingredients An efficiently implementable unitary matrix U on n qubits. Spectral decomposition U|ai = ei2πϕa |ai. k = polylog(n) auxiliary qubits in the state |0k i. Phase estimation maps |ai ⊗ |0k i → |ai ⊗ |ϕa i in poly time where X |ϕa i = ei2πϕa j |ji. j These are ’momentum’ states so the value of ϕa can be measured via Fourier transform. Substituting U = e−iHt with t < kHk−1 provides a method to measure the energy and hence to construct a projector onto H < α for use in Grover’s algorithm. David Poulin (Sherbrooke) Preparing Ground States CMS’08 10 / 19 The obvious method Phase estimation Ingredients An efficiently implementable unitary matrix U on n qubits. Spectral decomposition U|ai = ei2πϕa |ai. k = polylog(n) auxiliary qubits in the state |0k i. Phase estimation maps |ai ⊗ |0k i → |ai ⊗ |ϕa i in poly time where X |ϕa i = ei2πϕa j |ji. j These are ’momentum’ states so the value of ϕa can be measured via Fourier transform. Substituting U = e−iHt with t < kHk−1 provides a method to measure the energy and hence to construct a projector onto H < α for use in Grover’s algorithm. David Poulin (Sherbrooke) Preparing Ground States CMS’08 10 / 19 The obvious method Problem This is an approximate method to measure energy and errors build up during Grover’s algorithm. The unitary U = e−itH can only be approximated by the QC. Round-off errors in the Fourier transform. There is a small probability that an eigenstate |ai with phase ϕa << α be diagnoses as having a phase >> α. Let P be the projector onto the accepting subspace of the phase estimation algorithm. p √ |ai⊗|0k i = P|ai⊗|0k i+(1−P)|ai⊗|0k i = pa |gooda i+ 1 − pa |bada i Using Jordan’s block decomposition, we find that p √ |gooda i = pa |ai ⊗ |0k i − 1 − pa |ai ⊗ |0⊥ ki David Poulin (Sherbrooke) Preparing Ground States CMS’08 11 / 19 The obvious method Problem This is an approximate method to measure energy and errors build up during Grover’s algorithm. The unitary U = e−itH can only be approximated by the QC. Round-off errors in the Fourier transform. There is a small probability that an eigenstate |ai with phase ϕa << α be diagnoses as having a phase >> α. Let P be the projector onto the accepting subspace of the phase estimation algorithm. p √ |ai⊗|0k i = P|ai⊗|0k i+(1−P)|ai⊗|0k i = pa |gooda i+ 1 − pa |bada i Using Jordan’s block decomposition, we find that p √ |gooda i = pa |ai ⊗ |0k i − 1 − pa |ai ⊗ |0⊥ ki David Poulin (Sherbrooke) Preparing Ground States CMS’08 11 / 19 The obvious method Problem This is an approximate method to measure energy and errors build up during Grover’s algorithm. The unitary U = e−itH can only be approximated by the QC. Round-off errors in the Fourier transform. There is a small probability that an eigenstate |ai with phase ϕa << α be diagnoses as having a phase >> α. Let P be the projector onto the accepting subspace of the phase estimation algorithm. p √ |ai⊗|0k i = P|ai⊗|0k i+(1−P)|ai⊗|0k i = pa |gooda i+ 1 − pa |bada i Using Jordan’s block decomposition, we find that p √ |gooda i = pa |ai ⊗ |0k i − 1 − pa |ai ⊗ |0⊥ ki David Poulin (Sherbrooke) Preparing Ground States CMS’08 11 / 19 The obvious method Problem This is an approximate method to measure energy and errors build up during Grover’s algorithm. The unitary U = e−itH can only be approximated by the QC. Round-off errors in the Fourier transform. There is a small probability that an eigenstate |ai with phase ϕa << α be diagnoses as having a phase >> α. Let P be the projector onto the accepting subspace of the phase estimation algorithm. p √ |ai⊗|0k i = P|ai⊗|0k i+(1−P)|ai⊗|0k i = pa |gooda i+ 1 − pa |bada i Using Jordan’s block decomposition, we find that p √ |gooda i = pa |ai ⊗ |0k i − 1 − pa |ai ⊗ |0⊥ ki David Poulin (Sherbrooke) Preparing Ground States CMS’08 11 / 19 The obvious method Problem This is an approximate method to measure energy and errors build up during Grover’s algorithm. The unitary U = e−itH can only be approximated by the QC. Round-off errors in the Fourier transform. There is a small probability that an eigenstate |ai with phase ϕa << α be diagnoses as having a phase >> α. Let P be the projector onto the accepting subspace of the phase estimation algorithm. p √ |ai⊗|0k i = P|ai⊗|0k i+(1−P)|ai⊗|0k i = pa |gooda i+ 1 − pa |bada i Using Jordan’s block decomposition, we find that p √ |gooda i = pa |ai ⊗ |0k i − 1 − pa |ai ⊗ |0⊥ ki David Poulin (Sherbrooke) Preparing Ground States CMS’08 11 / 19 The obvious method Problem So let’s see what happens when we apply Grover’s algorithm to a random state... X |ψi = µa |ai ⊗ |0k i X √ p = µa pa |gooda i + 1 − pa |bada i 1 X √ Grover −−−−→ µa pa |gooda i kPψk p 1 X = µa (pa |ai ⊗ |0k i − pa (1 − pa )|ai ⊗ |0⊥ k i) kPψk We have entangled the system with the auxiliary qubits and the vast majority of the amplitude is on state with incorrect auxiliary qubits David Poulin (Sherbrooke) Preparing Ground States CMS’08 12 / 19 The obvious method Problem So let’s see what happens when we apply Grover’s algorithm to a random state... X |ψi = µa |ai ⊗ |0k i X √ p = µa pa |gooda i + 1 − pa |bada i 1 X √ Grover −−−−→ µa pa |gooda i kPψk p 1 X = µa (pa |ai ⊗ |0k i − pa (1 − pa )|ai ⊗ |0⊥ k i) kPψk We have entangled the system with the auxiliary qubits and the vast majority of the amplitude is on state with incorrect auxiliary qubits David Poulin (Sherbrooke) Preparing Ground States CMS’08 12 / 19 The obvious method Problem So let’s see what happens when we apply Grover’s algorithm to a random state... X |ψi = µa |ai ⊗ |0k i X √ p = µa pa |gooda i + 1 − pa |bada i 1 X √ Grover −−−−→ µa pa |gooda i kPψk p 1 X = µa (pa |ai ⊗ |0k i − pa (1 − pa )|ai ⊗ |0⊥ k i) kPψk We have entangled the system with the auxiliary qubits and the vast majority of the amplitude is on state with incorrect auxiliary qubits David Poulin (Sherbrooke) Preparing Ground States CMS’08 12 / 19 An algorithm to prepare ground states in time √ 2n Outline 1 Motivation 2 Existing methods 3 The obvious method 4 An algorithm to prepare ground states in time 5 Applications David Poulin (Sherbrooke) Preparing Ground States √ 2n CMS’08 13 / 19 An algorithm to prepare ground states in time √ 2n Running the algorithm backward Failed method P Prepare a µa |ai ⊗ |0k i. Use phase estimation to get P a µa |ai ⊗ |ϕa i. Amplify the low-momentum states ϕa ≤ α using an imperfect measurement. Better method P Prepare a µa |ai ⊗ |ωi for some ω ≤ α. P Use inverse phase estimation to get a µa hϕa |ωi|ai ⊗ |0k i + . . .. Amplifie the state |0k i of the auxiliary qubits. hϕa |ωi is a function of ϕa − ω peaked at 0 and of width 2−k = poly1(n) . This procedure thus filters out states whose energy is not near ω. David Poulin (Sherbrooke) Preparing Ground States CMS’08 14 / 19 An algorithm to prepare ground states in time √ 2n Running the algorithm backward Failed method P Prepare a µa |ai ⊗ |0k i. Use phase estimation to get P a µa |ai ⊗ |ϕa i. Amplify the low-momentum states ϕa ≤ α using an imperfect measurement. Better method P Prepare a µa |ai ⊗ |ωi for some ω ≤ α. P Use inverse phase estimation to get a µa hϕa |ωi|ai ⊗ |0k i + . . .. Amplifie the state |0k i of the auxiliary qubits. hϕa |ωi is a function of ϕa − ω peaked at 0 and of width 2−k = poly1(n) . This procedure thus filters out states whose energy is not near ω. David Poulin (Sherbrooke) Preparing Ground States CMS’08 14 / 19 An algorithm to prepare ground states in time √ 2n Running the algorithm backward Failed method P Prepare a µa |ai ⊗ |0k i. Use phase estimation to get P a µa |ai ⊗ |ϕa i. Amplify the low-momentum states ϕa ≤ α using an imperfect measurement. Better method P Prepare a µa |ai ⊗ |ωi for some ω ≤ α. P Use inverse phase estimation to get a µa hϕa |ωi|ai ⊗ |0k i + . . .. Amplifie the state |0k i of the auxiliary qubits. hϕa |ωi is a function of ϕa − ω peaked at 0 and of width 2−k = poly1(n) . This procedure thus filters out states whose energy is not near ω. David Poulin (Sherbrooke) Preparing Ground States CMS’08 14 / 19 An algorithm to prepare ground states in time √ 2n Running the algorithm backward Failed method P Prepare a µa |ai ⊗ |0k i. Use phase estimation to get P a µa |ai ⊗ |ϕa i. Amplify the low-momentum states ϕa ≤ α using an imperfect measurement. Better method P Prepare a µa |ai ⊗ |ωi for some ω ≤ α. P Use inverse phase estimation to get a µa hϕa |ωi|ai ⊗ |0k i + . . .. Amplifie the state |0k i of the auxiliary qubits. hϕa |ωi is a function of ϕa − ω peaked at 0 and of width 2−k = poly1(n) . This procedure thus filters out states whose energy is not near ω. David Poulin (Sherbrooke) Preparing Ground States CMS’08 14 / 19 An algorithm to prepare ground states in time √ 2n Running the algorithm backward Failed method P Prepare a µa |ai ⊗ |0k i. Use phase estimation to get P a µa |ai ⊗ |ϕa i. Amplify the low-momentum states ϕa ≤ α using an imperfect measurement. Better method P Prepare a µa |ai ⊗ |ωi for some ω ≤ α. P Use inverse phase estimation to get a µa hϕa |ωi|ai ⊗ |0k i + . . .. Amplifie the state |0k i of the auxiliary qubits. hϕa |ωi is a function of ϕa − ω peaked at 0 and of width 2−k = poly1(n) . This procedure thus filters out states whose energy is not near ω. David Poulin (Sherbrooke) Preparing Ground States CMS’08 14 / 19 An algorithm to prepare ground states in time √ 2n Running the algorithm backward Failed method P Prepare a µa |ai ⊗ |0k i. Use phase estimation to get P a µa |ai ⊗ |ϕa i. Amplify the low-momentum states ϕa ≤ α using an imperfect measurement. Better method P Prepare a µa |ai ⊗ |ωi for some ω ≤ α. P Use inverse phase estimation to get a µa hϕa |ωi|ai ⊗ |0k i + . . .. Amplifie the state |0k i of the auxiliary qubits. hϕa |ωi is a function of ϕa − ω peaked at 0 and of width 2−k = poly1(n) . This procedure thus filters out states whose energy is not near ω. David Poulin (Sherbrooke) Preparing Ground States CMS’08 14 / 19 An algorithm to prepare ground states in time √ 2n A better filter Our filter has a heavy tail: |hϕa |ωi| ≤ 1/(2k +1 |ϕa − ω|) = 1/(poly (n)|ϕa − ω|). Since there are an exponential number of states that can contribute to the energy, we need an exponential filter. Apply the filter η ≈ n times: |hϕa |ωi| → |hϕa |ωi|η ≈ |ϕa − ω|−η With k ≥ 2 log2 ( 1 ) and η ≥ 1 + (n + 1)/ log2 ( 1 ) we can show: When there is no eigenvalue of H in the interval ω ± /2, Grover’s algorithm succeeds with probability ≤ 2−n . √ If a state lies in the interval ω ± 2−k /(2π η), Grover’s algorithm has a constant probability of succeeding. When Grover’s algorithm succeeds, its output has energy within ω ± /2 with probability ≥ 1 − 2−n . David Poulin (Sherbrooke) Preparing Ground States CMS’08 15 / 19 An algorithm to prepare ground states in time √ 2n A better filter Our filter has a heavy tail: |hϕa |ωi| ≤ 1/(2k +1 |ϕa − ω|) = 1/(poly (n)|ϕa − ω|). Since there are an exponential number of states that can contribute to the energy, we need an exponential filter. Apply the filter η ≈ n times: |hϕa |ωi| → |hϕa |ωi|η ≈ |ϕa − ω|−η With k ≥ 2 log2 ( 1 ) and η ≥ 1 + (n + 1)/ log2 ( 1 ) we can show: When there is no eigenvalue of H in the interval ω ± /2, Grover’s algorithm succeeds with probability ≤ 2−n . √ If a state lies in the interval ω ± 2−k /(2π η), Grover’s algorithm has a constant probability of succeeding. When Grover’s algorithm succeeds, its output has energy within ω ± /2 with probability ≥ 1 − 2−n . David Poulin (Sherbrooke) Preparing Ground States CMS’08 15 / 19 An algorithm to prepare ground states in time √ 2n A better filter Our filter has a heavy tail: |hϕa |ωi| ≤ 1/(2k +1 |ϕa − ω|) = 1/(poly (n)|ϕa − ω|). Since there are an exponential number of states that can contribute to the energy, we need an exponential filter. Apply the filter η ≈ n times: |hϕa |ωi| → |hϕa |ωi|η ≈ |ϕa − ω|−η With k ≥ 2 log2 ( 1 ) and η ≥ 1 + (n + 1)/ log2 ( 1 ) we can show: When there is no eigenvalue of H in the interval ω ± /2, Grover’s algorithm succeeds with probability ≤ 2−n . √ If a state lies in the interval ω ± 2−k /(2π η), Grover’s algorithm has a constant probability of succeeding. When Grover’s algorithm succeeds, its output has energy within ω ± /2 with probability ≥ 1 − 2−n . David Poulin (Sherbrooke) Preparing Ground States CMS’08 15 / 19 An algorithm to prepare ground states in time √ 2n A better filter Our filter has a heavy tail: |hϕa |ωi| ≤ 1/(2k +1 |ϕa − ω|) = 1/(poly (n)|ϕa − ω|). Since there are an exponential number of states that can contribute to the energy, we need an exponential filter. Apply the filter η ≈ n times: |hϕa |ωi| → |hϕa |ωi|η ≈ |ϕa − ω|−η With k ≥ 2 log2 ( 1 ) and η ≥ 1 + (n + 1)/ log2 ( 1 ) we can show: When there is no eigenvalue of H in the interval ω ± /2, Grover’s algorithm succeeds with probability ≤ 2−n . √ If a state lies in the interval ω ± 2−k /(2π η), Grover’s algorithm has a constant probability of succeeding. When Grover’s algorithm succeeds, its output has energy within ω ± /2 with probability ≥ 1 − 2−n . David Poulin (Sherbrooke) Preparing Ground States CMS’08 15 / 19 An algorithm to prepare ground states in time √ 2n A better filter Our filter has a heavy tail: |hϕa |ωi| ≤ 1/(2k +1 |ϕa − ω|) = 1/(poly (n)|ϕa − ω|). Since there are an exponential number of states that can contribute to the energy, we need an exponential filter. Apply the filter η ≈ n times: |hϕa |ωi| → |hϕa |ωi|η ≈ |ϕa − ω|−η With k ≥ 2 log2 ( 1 ) and η ≥ 1 + (n + 1)/ log2 ( 1 ) we can show: When there is no eigenvalue of H in the interval ω ± /2, Grover’s algorithm succeeds with probability ≤ 2−n . √ If a state lies in the interval ω ± 2−k /(2π η), Grover’s algorithm has a constant probability of succeeding. When Grover’s algorithm succeeds, its output has energy within ω ± /2 with probability ≥ 1 − 2−n . David Poulin (Sherbrooke) Preparing Ground States CMS’08 15 / 19 An algorithm to prepare ground states in time √ 2n A better filter Our filter has a heavy tail: |hϕa |ωi| ≤ 1/(2k +1 |ϕa − ω|) = 1/(poly (n)|ϕa − ω|). Since there are an exponential number of states that can contribute to the energy, we need an exponential filter. Apply the filter η ≈ n times: |hϕa |ωi| → |hϕa |ωi|η ≈ |ϕa − ω|−η With k ≥ 2 log2 ( 1 ) and η ≥ 1 + (n + 1)/ log2 ( 1 ) we can show: When there is no eigenvalue of H in the interval ω ± /2, Grover’s algorithm succeeds with probability ≤ 2−n . √ If a state lies in the interval ω ± 2−k /(2π η), Grover’s algorithm has a constant probability of succeeding. When Grover’s algorithm succeeds, its output has energy within ω ± /2 with probability ≥ 1 − 2−n . David Poulin (Sherbrooke) Preparing Ground States CMS’08 15 / 19 Applications Outline 1 Motivation 2 Existing methods 3 The obvious method 4 An algorithm to prepare ground states in time 5 Applications David Poulin (Sherbrooke) Preparing Ground States √ 2n CMS’08 16 / 19 Applications Preparing thermal states Thermal state at temperature T is 1 −H/T Ze where Z = Tr {e−H/T }. Quantum counting combined P with previous method to estimate density of state D(ω) = a δ(ω − ϕa ). Choose a random ω following the distribution e−ω/T D(ω). Use the previous algorithm to create a state of energy near ω. David Poulin (Sherbrooke) Preparing Ground States CMS’08 17 / 19 Applications Preparing thermal states Thermal state at temperature T is 1 −H/T Ze where Z = Tr {e−H/T }. Quantum counting combined P with previous method to estimate density of state D(ω) = a δ(ω − ϕa ). Choose a random ω following the distribution e−ω/T D(ω). Use the previous algorithm to create a state of energy near ω. David Poulin (Sherbrooke) Preparing Ground States CMS’08 17 / 19 Applications Preparing thermal states Thermal state at temperature T is 1 −H/T Ze where Z = Tr {e−H/T }. Quantum counting combined P with previous method to estimate density of state D(ω) = a δ(ω − ϕa ). Choose a random ω following the distribution e−ω/T D(ω). Use the previous algorithm to create a state of energy near ω. David Poulin (Sherbrooke) Preparing Ground States CMS’08 17 / 19 Applications Preparing thermal states Thermal state at temperature T is 1 −H/T Ze where Z = Tr {e−H/T }. Quantum counting combined P with previous method to estimate density of state D(ω) = a δ(ω − ϕa ). Choose a random ω following the distribution e−ω/T D(ω). Use the previous algorithm to create a state of energy near ω. David Poulin (Sherbrooke) Preparing Ground States CMS’08 17 / 19 Applications Solving QMA Given an efficient quantum circuit V , maximize the probability of observing its first output qubit in the state 1 over all input states if the form |ψn i ⊗ |0h i. Reduces to local Hamiltonian (Kiatev) but this does not preserve witness size. Acceptance probability of V ⇔ Energy of H. Marriott and Watrous have designed a circuit to estimate the acceptance probability of V : David Poulin (Sherbrooke) Preparing Ground States CMS’08 18 / 19 Applications Solving QMA Given an efficient quantum circuit V , maximize the probability of observing its first output qubit in the state 1 over all input states if the form |ψn i ⊗ |0h i. Reduces to local Hamiltonian (Kiatev) but this does not preserve witness size. Acceptance probability of V ⇔ Energy of H. Marriott and Watrous have designed a circuit to estimate the acceptance probability of V : David Poulin (Sherbrooke) Preparing Ground States CMS’08 18 / 19 Applications Solving QMA Given an efficient quantum circuit V , maximize the probability of observing its first output qubit in the state 1 over all input states if the form |ψn i ⊗ |0h i. Reduces to local Hamiltonian (Kiatev) but this does not preserve witness size. Acceptance probability of V ⇔ Energy of H. Marriott and Watrous have designed a circuit to estimate the acceptance probability of V : David Poulin (Sherbrooke) Preparing Ground States CMS’08 18 / 19 Applications Solving QMA Given an efficient quantum circuit V , maximize the probability of observing its first output qubit in the state 1 over all input states if the form |ψn i ⊗ |0h i. Reduces to local Hamiltonian (Kiatev) but this does not preserve witness size. Acceptance probability of V ⇔ Energy of H. Marriott and Watrous have designed a circuit to estimate the acceptance probability of V : V V -1 V -1 V V -1 V |ωi = David Poulin (Sherbrooke) P √ j ω Preparing Ground States k −s(w) √ 1−ω s(j) (−1)`(j) |ji CMS’08 18 / 19 Summary Conclusion For a quantum computer... Preparing the ground state of a quantum system is no more difficult than preparing the ground state of a classical system. QMA is no harder than NP† . † Assuming that all problems in QMA require only a log-size scratch pad. David Poulin (Sherbrooke) Preparing Ground States CMS’08 19 / 19