Problem with the mathematical formulation of “qubitization”












5












$begingroup$


In this research paper, the authors introduce a new algorithm to perform Hamiltonian simulation.



The beginning of their abstract is




Given a Hermitian operator $hat{H} = langle Gvert hat{U} vert Grangle$ that is the projection of an oracle $hat{U}$ by state $vert Grangle$
created with oracle $hat{G}$, the problem of Hamiltonian simulation is approximating the time evolution operator $e^{-ihat{H}t}$ at time $t$ with error $epsilon$.




In the article:





  • $hat{G}$ and $hat{U}$ are called "oracles".


  • $hat{H}$ is an Hermitian operator in $mathbb{C}^{2^n} times mathbb{C}^{2^n}$.


  • $vert G rangle in mathbb{C}^d$ (legend of Table 1).


My question is the following: what means $hat{H} = langle Gvert hat{U} vert Grangle$? More precisely, I do not understand what $langle Gvert hat{U} vert Grangle$ represents when $hat{U}$ is an oracle and $vert G rangle$ a quantum state.






mathematics hamiltonian-simulation notation







share|improve this question











$endgroup$

















    5












    $begingroup$


    In this research paper, the authors introduce a new algorithm to perform Hamiltonian simulation.



    The beginning of their abstract is




    Given a Hermitian operator $hat{H} = langle Gvert hat{U} vert Grangle$ that is the projection of an oracle $hat{U}$ by state $vert Grangle$
    created with oracle $hat{G}$, the problem of Hamiltonian simulation is approximating the time evolution operator $e^{-ihat{H}t}$ at time $t$ with error $epsilon$.




    In the article:





    • $hat{G}$ and $hat{U}$ are called "oracles".


    • $hat{H}$ is an Hermitian operator in $mathbb{C}^{2^n} times mathbb{C}^{2^n}$.


    • $vert G rangle in mathbb{C}^d$ (legend of Table 1).


    My question is the following: what means $hat{H} = langle Gvert hat{U} vert Grangle$? More precisely, I do not understand what $langle Gvert hat{U} vert Grangle$ represents when $hat{U}$ is an oracle and $vert G rangle$ a quantum state.






    mathematics hamiltonian-simulation notation







    share|improve this question











    $endgroup$















      5












      5








      5


      1



      $begingroup$


      In this research paper, the authors introduce a new algorithm to perform Hamiltonian simulation.



      The beginning of their abstract is




      Given a Hermitian operator $hat{H} = langle Gvert hat{U} vert Grangle$ that is the projection of an oracle $hat{U}$ by state $vert Grangle$
      created with oracle $hat{G}$, the problem of Hamiltonian simulation is approximating the time evolution operator $e^{-ihat{H}t}$ at time $t$ with error $epsilon$.




      In the article:





      • $hat{G}$ and $hat{U}$ are called "oracles".


      • $hat{H}$ is an Hermitian operator in $mathbb{C}^{2^n} times mathbb{C}^{2^n}$.


      • $vert G rangle in mathbb{C}^d$ (legend of Table 1).


      My question is the following: what means $hat{H} = langle Gvert hat{U} vert Grangle$? More precisely, I do not understand what $langle Gvert hat{U} vert Grangle$ represents when $hat{U}$ is an oracle and $vert G rangle$ a quantum state.






      mathematics hamiltonian-simulation notation







      share|improve this question











      $endgroup$




      In this research paper, the authors introduce a new algorithm to perform Hamiltonian simulation.



      The beginning of their abstract is




      Given a Hermitian operator $hat{H} = langle Gvert hat{U} vert Grangle$ that is the projection of an oracle $hat{U}$ by state $vert Grangle$
      created with oracle $hat{G}$, the problem of Hamiltonian simulation is approximating the time evolution operator $e^{-ihat{H}t}$ at time $t$ with error $epsilon$.




      In the article:





      • $hat{G}$ and $hat{U}$ are called "oracles".


      • $hat{H}$ is an Hermitian operator in $mathbb{C}^{2^n} times mathbb{C}^{2^n}$.


      • $vert G rangle in mathbb{C}^d$ (legend of Table 1).


      My question is the following: what means $hat{H} = langle Gvert hat{U} vert Grangle$? More precisely, I do not understand what $langle Gvert hat{U} vert Grangle$ represents when $hat{U}$ is an oracle and $vert G rangle$ a quantum state.






      mathematics hamiltonian-simulation notation




      mathematics hamiltonian-simulation notation






      share|improve this question















      share|improve this question













      share|improve this question




      share|improve this question








      edited Dec 11 '18 at 13:33







      Nelimee

















      asked Dec 11 '18 at 12:58









      NelimeeNelimee

      1,570326




      1,570326






















          1 Answer
          1






          active

          oldest

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          4












          $begingroup$

          You want to start by being careful with the sizes of the operators. $hat U$ acts on $q$ qubits, and $hat H$ acts on $n<q$ qubits. I believe that $|Grangle$ is a state of $q-n$ qubits. So, what we really need to talk about is two distinct sets of qubits. Let me call them sets $A$ and $B$. $A$ contains $n$ qubits, and $B$ contains $q-n$ qubits. I'll use subscripts to denote which qubits the different operators and states act upon:



          $$
          hat H_A=(langle G|_Botimesmathbb{I}_A)hat U_{AB}(|Grangle_Botimesmathbb{I}_A)
          $$






          share|improve this answer









          $endgroup$













          • $begingroup$
            @Nelimee I'm not sure if this is sufficient to resolve your confusion? Or is there something more that you're asking?
            $endgroup$
            – DaftWullie
            Dec 11 '18 at 13:24










          • $begingroup$
            I am still trying to understand your answer but the sizes of the operators were definitely one of the points I missed! About your answer, what does $vert G rangle_B otimes mathbb{I}_A$ represent? A tensor product between a quantum state (a vector) and an operator (a matrix)?
            $endgroup$
            – Nelimee
            Dec 11 '18 at 13:39










          • $begingroup$
            Yes, exactly. Where, of course, you should think of a vector as a matrix where one of the dimensions is just 1.
            $endgroup$
            – DaftWullie
            Dec 11 '18 at 13:45










          • $begingroup$
            Ok that solved my problem! Thanks for the quick clarification :)
            $endgroup$
            – Nelimee
            Dec 11 '18 at 13:54











          Your Answer





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          1 Answer
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          active

          oldest

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          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

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          active

          oldest

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          4












          $begingroup$

          You want to start by being careful with the sizes of the operators. $hat U$ acts on $q$ qubits, and $hat H$ acts on $n<q$ qubits. I believe that $|Grangle$ is a state of $q-n$ qubits. So, what we really need to talk about is two distinct sets of qubits. Let me call them sets $A$ and $B$. $A$ contains $n$ qubits, and $B$ contains $q-n$ qubits. I'll use subscripts to denote which qubits the different operators and states act upon:



          $$
          hat H_A=(langle G|_Botimesmathbb{I}_A)hat U_{AB}(|Grangle_Botimesmathbb{I}_A)
          $$






          share|improve this answer









          $endgroup$













          • $begingroup$
            @Nelimee I'm not sure if this is sufficient to resolve your confusion? Or is there something more that you're asking?
            $endgroup$
            – DaftWullie
            Dec 11 '18 at 13:24










          • $begingroup$
            I am still trying to understand your answer but the sizes of the operators were definitely one of the points I missed! About your answer, what does $vert G rangle_B otimes mathbb{I}_A$ represent? A tensor product between a quantum state (a vector) and an operator (a matrix)?
            $endgroup$
            – Nelimee
            Dec 11 '18 at 13:39










          • $begingroup$
            Yes, exactly. Where, of course, you should think of a vector as a matrix where one of the dimensions is just 1.
            $endgroup$
            – DaftWullie
            Dec 11 '18 at 13:45










          • $begingroup$
            Ok that solved my problem! Thanks for the quick clarification :)
            $endgroup$
            – Nelimee
            Dec 11 '18 at 13:54
















          4












          $begingroup$

          You want to start by being careful with the sizes of the operators. $hat U$ acts on $q$ qubits, and $hat H$ acts on $n<q$ qubits. I believe that $|Grangle$ is a state of $q-n$ qubits. So, what we really need to talk about is two distinct sets of qubits. Let me call them sets $A$ and $B$. $A$ contains $n$ qubits, and $B$ contains $q-n$ qubits. I'll use subscripts to denote which qubits the different operators and states act upon:



          $$
          hat H_A=(langle G|_Botimesmathbb{I}_A)hat U_{AB}(|Grangle_Botimesmathbb{I}_A)
          $$






          share|improve this answer









          $endgroup$













          • $begingroup$
            @Nelimee I'm not sure if this is sufficient to resolve your confusion? Or is there something more that you're asking?
            $endgroup$
            – DaftWullie
            Dec 11 '18 at 13:24










          • $begingroup$
            I am still trying to understand your answer but the sizes of the operators were definitely one of the points I missed! About your answer, what does $vert G rangle_B otimes mathbb{I}_A$ represent? A tensor product between a quantum state (a vector) and an operator (a matrix)?
            $endgroup$
            – Nelimee
            Dec 11 '18 at 13:39










          • $begingroup$
            Yes, exactly. Where, of course, you should think of a vector as a matrix where one of the dimensions is just 1.
            $endgroup$
            – DaftWullie
            Dec 11 '18 at 13:45










          • $begingroup$
            Ok that solved my problem! Thanks for the quick clarification :)
            $endgroup$
            – Nelimee
            Dec 11 '18 at 13:54














          4












          4








          4





          $begingroup$

          You want to start by being careful with the sizes of the operators. $hat U$ acts on $q$ qubits, and $hat H$ acts on $n<q$ qubits. I believe that $|Grangle$ is a state of $q-n$ qubits. So, what we really need to talk about is two distinct sets of qubits. Let me call them sets $A$ and $B$. $A$ contains $n$ qubits, and $B$ contains $q-n$ qubits. I'll use subscripts to denote which qubits the different operators and states act upon:



          $$
          hat H_A=(langle G|_Botimesmathbb{I}_A)hat U_{AB}(|Grangle_Botimesmathbb{I}_A)
          $$






          share|improve this answer









          $endgroup$



          You want to start by being careful with the sizes of the operators. $hat U$ acts on $q$ qubits, and $hat H$ acts on $n<q$ qubits. I believe that $|Grangle$ is a state of $q-n$ qubits. So, what we really need to talk about is two distinct sets of qubits. Let me call them sets $A$ and $B$. $A$ contains $n$ qubits, and $B$ contains $q-n$ qubits. I'll use subscripts to denote which qubits the different operators and states act upon:



          $$
          hat H_A=(langle G|_Botimesmathbb{I}_A)hat U_{AB}(|Grangle_Botimesmathbb{I}_A)
          $$







          share|improve this answer












          share|improve this answer



          share|improve this answer










          answered Dec 11 '18 at 13:23









          DaftWullieDaftWullie

          13.9k1540




          13.9k1540












          • $begingroup$
            @Nelimee I'm not sure if this is sufficient to resolve your confusion? Or is there something more that you're asking?
            $endgroup$
            – DaftWullie
            Dec 11 '18 at 13:24










          • $begingroup$
            I am still trying to understand your answer but the sizes of the operators were definitely one of the points I missed! About your answer, what does $vert G rangle_B otimes mathbb{I}_A$ represent? A tensor product between a quantum state (a vector) and an operator (a matrix)?
            $endgroup$
            – Nelimee
            Dec 11 '18 at 13:39










          • $begingroup$
            Yes, exactly. Where, of course, you should think of a vector as a matrix where one of the dimensions is just 1.
            $endgroup$
            – DaftWullie
            Dec 11 '18 at 13:45










          • $begingroup$
            Ok that solved my problem! Thanks for the quick clarification :)
            $endgroup$
            – Nelimee
            Dec 11 '18 at 13:54


















          • $begingroup$
            @Nelimee I'm not sure if this is sufficient to resolve your confusion? Or is there something more that you're asking?
            $endgroup$
            – DaftWullie
            Dec 11 '18 at 13:24










          • $begingroup$
            I am still trying to understand your answer but the sizes of the operators were definitely one of the points I missed! About your answer, what does $vert G rangle_B otimes mathbb{I}_A$ represent? A tensor product between a quantum state (a vector) and an operator (a matrix)?
            $endgroup$
            – Nelimee
            Dec 11 '18 at 13:39










          • $begingroup$
            Yes, exactly. Where, of course, you should think of a vector as a matrix where one of the dimensions is just 1.
            $endgroup$
            – DaftWullie
            Dec 11 '18 at 13:45










          • $begingroup$
            Ok that solved my problem! Thanks for the quick clarification :)
            $endgroup$
            – Nelimee
            Dec 11 '18 at 13:54
















          $begingroup$
          @Nelimee I'm not sure if this is sufficient to resolve your confusion? Or is there something more that you're asking?
          $endgroup$
          – DaftWullie
          Dec 11 '18 at 13:24




          $begingroup$
          @Nelimee I'm not sure if this is sufficient to resolve your confusion? Or is there something more that you're asking?
          $endgroup$
          – DaftWullie
          Dec 11 '18 at 13:24












          $begingroup$
          I am still trying to understand your answer but the sizes of the operators were definitely one of the points I missed! About your answer, what does $vert G rangle_B otimes mathbb{I}_A$ represent? A tensor product between a quantum state (a vector) and an operator (a matrix)?
          $endgroup$
          – Nelimee
          Dec 11 '18 at 13:39




          $begingroup$
          I am still trying to understand your answer but the sizes of the operators were definitely one of the points I missed! About your answer, what does $vert G rangle_B otimes mathbb{I}_A$ represent? A tensor product between a quantum state (a vector) and an operator (a matrix)?
          $endgroup$
          – Nelimee
          Dec 11 '18 at 13:39












          $begingroup$
          Yes, exactly. Where, of course, you should think of a vector as a matrix where one of the dimensions is just 1.
          $endgroup$
          – DaftWullie
          Dec 11 '18 at 13:45




          $begingroup$
          Yes, exactly. Where, of course, you should think of a vector as a matrix where one of the dimensions is just 1.
          $endgroup$
          – DaftWullie
          Dec 11 '18 at 13:45












          $begingroup$
          Ok that solved my problem! Thanks for the quick clarification :)
          $endgroup$
          – Nelimee
          Dec 11 '18 at 13:54




          $begingroup$
          Ok that solved my problem! Thanks for the quick clarification :)
          $endgroup$
          – Nelimee
          Dec 11 '18 at 13:54


















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