Problem with the mathematical formulation of “qubitization”











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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.










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    up vote
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    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.










    share|improve this question


























      up vote
      5
      down vote

      favorite









      up vote
      5
      down vote

      favorite











      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.










      share|improve this question















      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






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      edited Dec 11 at 13:33

























      asked Dec 11 at 12:58









      Nelimee

      1,270225




      1,270225






















          1 Answer
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          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





















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










          • 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)?
            – Nelimee
            Dec 11 at 13:39










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










          • Ok that solved my problem! Thanks for the quick clarification :)
            – Nelimee
            Dec 11 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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          up vote
          4
          down vote



          accepted










          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





















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










          • 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)?
            – Nelimee
            Dec 11 at 13:39










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










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















          up vote
          4
          down vote



          accepted










          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





















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










          • 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)?
            – Nelimee
            Dec 11 at 13:39










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










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













          up vote
          4
          down vote



          accepted







          up vote
          4
          down vote



          accepted






          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












          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 at 13:23









          DaftWullie

          11.6k1536




          11.6k1536












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










          • 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)?
            – Nelimee
            Dec 11 at 13:39










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










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


















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










          • 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)?
            – Nelimee
            Dec 11 at 13:39










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










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
















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




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












          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)?
          – Nelimee
          Dec 11 at 13:39




          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)?
          – Nelimee
          Dec 11 at 13:39












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




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












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




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


















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