<p>To realize correct quantum computing it seems that both quantum error correction and quantum tomography would be necessary. Is this true? What is the relationship between these two fields?</p>

CC BY-SA 3.0
2015-11-03T04:07:26.757Z
Yes, this is probably true. And you also have to realize quantum fault-tolerance. Quantum error correction is storing or transmitting quantum information in a manner that lets you correct errors in it. Quantum tomography is figuring out what a quantum process is actually doing. And even if in theory you know what your quantum gates do, you probably need to check them using quantum tomography. Quantum fault tolerance is correcting errors on a quantum computer faster than they occur. These are connected in a several ways—e.g., similar techniques.
Peter Shor 
CC BY-SA 3.0
2015-11-03T15:00:10.733Z
If I understand correctly, Wiki on quantum tomography claims that for more than a few quantum bits quantum tomography is impossible in practice owing to exponential increase of work. Wouldn't that imply that the chance of having a practical quantum computer is extremely low from the very beginning?
Mok-Kong Shen
CC BY-SA 3.0
2015-11-06T09:01:24.020Z
@PeterShor: I should very much appreciate it, if you would with your expertise also kindly help in the question I raised in my last comment.
Mok-Kong Shen
CC BY-SA 3.0
2015-11-06T10:47:21.377Z
The reason fault tolerance works on a quantum computer is that the effect of quantum gates is confined to only a few qubits. Thus, you only need to use quantum tomography on a few qubits to figure out what your gates are doing and what the error model for them should be.
Peter Shor 
CC BY-SA 3.0
2015-11-07T09:02:21.020Z
@PeterShor: But a practical quantum computer has presumably a fairly sizeable number of gates and, while design is mostly a onetime work, the work involved in manufacture and maintenance might constitute a formidable issue in practice, I would as layman surmise.
Mok-Kong Shen
CC BY-SA 3.0
2015-11-07T15:25:44.890Z
[Addendum to my last comment:] A circuit may contain a number of (basic) gates. While for the classical hardware the individual gates may be arbitrarily separately tested at different time points, this wouldn't be feasible, I am afraid, for quantum hardware, i.e. the process implemented by the whole circuit would have to be verified as a single job. (I surmise that this is at the base of the exponential increase of work with the number of qubits as claimed in the Wiki article on quantum process tomography.)
Mok-Kong Shen
CC BY-SA 3.0
2015-11-07T16:18:06.233Z
Why don't you think you can test a single gate that is part of a quantum computer? If the test was performed in the middle of a quantum computation, it might affect the results. But before the computation, why not?
Peter Shor 
CC BY-SA 3.0
2015-11-08T10:42:38.927Z
@PeterShor: The correctness of the "combined" operation of a number of gates in a circuit (inputs vs. outputs of the circuit!)  is at issue. In classical hardware one can arbitrarily choose test points and their time ordering in the circuit, but for quantum hardware one test destroys the qubit there and  would thus affect the test result for the whole circuit, as you wrote. As layman I surmise that it is principally because of this that QPT relies on statistical tests of the entire process, which means large numbers of tests and explains the exponential increase of work and also the P in QPT.
Mok-Kong Shen
CC BY-SA 3.0
2015-11-09T11:20:05.623Z
@PeterShor: I currently suppose (i.e. pending eventual contradictions from you) that the field of quantum tomography would play a decisive role in the hoped-for success of future practical quantum computing. What I find it difficult to understand is though that the some 10 books on quantum computing I have looked at don't even mention quantum tomography and that a news article I happend to read recently went so far as to say that a certain newly achieved technical breakthrough would enable (regular) production of quantum chips in a few years, in case financiall sponsorship could be available.
Mok-Kong Shen
CC BY-SA 3.0
2015-11-09T11:57:53.663Z
The basic idea is that if your quantum gates work reasonably well, and your qubits and gates all behave reasonably independently, then your quantum computer will give the correct result. This is the fault-tolerance theorem. I am going to stop trying to explain things in these comments. I don't have enough time to give detailed explanations, the fault-tolerance theorem is way too complicated for a detailed, short explanation, and it also seems to me that you are not actually listening to what I'm saying.
Peter Shor 
CC BY-SA 3.0
2015-11-09T15:07:43.797Z
@PeterShor: I did pay attention to what you wrote. But I am possibly highly influenced by my work in classical computing where the hardware needs verification on diverse occassions, namely design, manufacture and maintenance. Actually (1) recently I asked a physicist how one could know that a quantum gate (note not even a circuit) works as specified and he referred me to quantum tomography (2) elsewhere in a discussion about hardware I mentioned that classical hardware has constantly checks while running and that decades ago mainframes had maintenance times of a couple of hours or more daily.
Mok-Kong Shen
CC BY-SA 3.0
2015-11-09T15:19:44.787Z
[Addendum due to length limit of comment: ] Correct functioning of a quantum computer would certainly depend on appropriate runtime checks and maintenance, exactly the same as in the case of classical computers. Thus, if that's too costly or practically infeasible, quantum computing's future would be highly questionable IMHO, despite the enormous research work being done today.
Mok-Kong Shen
CC BY-SA 3.0
2015-11-11T05:16:52.940Z
You ***could*** build classical hardware without run-time checks and have it work; it's just cheaper and more efficient to use run-time checks. For quantum computers, you instead have to use fault-tolerance techniques that are not used on classical computers because the overhead is a lot higher than run-time checks. But it's not exponential, the way you seem to think it would be.
Peter Shor 
CC BY-SA 3.0
2015-11-11T09:47:36.547Z
@PeterShor: Verification in my view subsumes error correction. Error correction deals with unvoidable noises, while verification asks whether a circuit or gate as such (i.e.  even in cases there is no noise) at the test time delivers results as specified (problems could be due to design mistakes, faults that occur in the manufacturing process and wrong handling by maintenance personal and these are clearly not in the realm of error correction). Thus In my understanding quantum process tomography is necessary in combination with error correction to achieve correct quantum computing in practice.
Mok-Kong Shen
<quantum-computing>
What is the relationship between quantum tomography and quantum error correction?
Mok-Kong Shen
# What is the relationship between quantum tomography and quantum error correction?

(Asked by: username_0 on 2015-10-29)

To realize correct quantum computing it seems that both quantum error correction and quantum tomography would be necessary. Is this true? What is the relationship between these two fields?

**Tags:** quantum-computing

> **Comments:**
> **username_1**: Yes, this is probably true. And you also have to realize quantum fault-tolerance. Quantum error correction is storing or transmitting quantum information in a manner that lets you correct errors in it. Quantum tomography is figuring out what a quantum process is actually doing. And even if in theory you know what your quantum gates do, you probably need to check them using quantum tomography. Quantum fault tolerance is correcting errors on a quantum computer faster than they occur. These are connected in a several ways—e.g., similar techniques.  
> **username_0**: If I understand correctly, Wiki on quantum tomography claims that for more than a few quantum bits quantum tomography is impossible in practice owing to exponential increase of work. Wouldn't that imply that the chance of having a practical quantum computer is extremely low from the very beginning?  
> **username_0**: @PeterShor: I should very much appreciate it, if you would with your expertise also kindly help in the question I raised in my last comment.  
> **username_1**: The reason fault tolerance works on a quantum computer is that the effect of quantum gates is confined to only a few qubits. Thus, you only need to use quantum tomography on a few qubits to figure out what your gates are doing and what the error model for them should be.  
> **username_0**: @PeterShor: But a practical quantum computer has presumably a fairly sizeable number of gates and, while design is mostly a onetime work, the work involved in manufacture and maintenance might constitute a formidable issue in practice, I would as layman surmise.  
> **username_0**: [Addendum to my last comment:] A circuit may contain a number of (basic) gates. While for the classical hardware the individual gates may be arbitrarily separately tested at different time points, this wouldn't be feasible, I am afraid, for quantum hardware, i.e. the process implemented by the whole circuit would have to be verified as a single job. (I surmise that this is at the base of the exponential increase of work with the number of qubits as claimed in the Wiki article on quantum process tomography.)  
> **username_1**: Why don't you think you can test a single gate that is part of a quantum computer? If the test was performed in the middle of a quantum computation, it might affect the results. But before the computation, why not?  
> **username_0**: @PeterShor: The correctness of the "combined" operation of a number of gates in a circuit (inputs vs. outputs of the circuit!)  is at issue. In classical hardware one can arbitrarily choose test points and their time ordering in the circuit, but for quantum hardware one test destroys the qubit there and  would thus affect the test result for the whole circuit, as you wrote. As layman I surmise that it is principally because of this that QPT relies on statistical tests of the entire process, which means large numbers of tests and explains the exponential increase of work and also the P in QPT.  
> **username_0**: @PeterShor: I currently suppose (i.e. pending eventual contradictions from you) that the field of quantum tomography would play a decisive role in the hoped-for success of future practical quantum computing. What I find it difficult to understand is though that the some 10 books on quantum computing I have looked at don't even mention quantum tomography and that a news article I happend to read recently went so far as to say that a certain newly achieved technical breakthrough would enable (regular) production of quantum chips in a few years, in case financiall sponsorship could be available.  
> **username_1**: The basic idea is that if your quantum gates work reasonably well, and your qubits and gates all behave reasonably independently, then your quantum computer will give the correct result. This is the fault-tolerance theorem. I am going to stop trying to explain things in these comments. I don't have enough time to give detailed explanations, the fault-tolerance theorem is way too complicated for a detailed, short explanation, and it also seems to me that you are not actually listening to what I'm saying.  
> **username_0**: @PeterShor: I did pay attention to what you wrote. But I am possibly highly influenced by my work in classical computing where the hardware needs verification on diverse occassions, namely design, manufacture and maintenance. Actually (1) recently I asked a physicist how one could know that a quantum gate (note not even a circuit) works as specified and he referred me to quantum tomography (2) elsewhere in a discussion about hardware I mentioned that classical hardware has constantly checks while running and that decades ago mainframes had maintenance times of a couple of hours or more daily.  
> **username_0**: [Addendum due to length limit of comment: ] Correct functioning of a quantum computer would certainly depend on appropriate runtime checks and maintenance, exactly the same as in the case of classical computers. Thus, if that's too costly or practically infeasible, quantum computing's future would be highly questionable IMHO, despite the enormous research work being done today.  
> **username_1**: You ***could*** build classical hardware without run-time checks and have it work; it's just cheaper and more efficient to use run-time checks. For quantum computers, you instead have to use fault-tolerance techniques that are not used on classical computers because the overhead is a lot higher than run-time checks. But it's not exponential, the way you seem to think it would be.  
> **username_0**: @PeterShor: Verification in my view subsumes error correction. Error correction deals with unvoidable noises, while verification asks whether a circuit or gate as such (i.e.  even in cases there is no noise) at the test time delivers results as specified (problems could be due to design mistakes, faults that occur in the manufacturing process and wrong handling by maintenance personal and these are clearly not in the realm of error correction). Thus In my understanding quantum process tomography is necessary in combination with error correction to achieve correct quantum computing in practice.  

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## 0 answers

