Difference between revisions of "Research:Self Correcting Quantum Computers"
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Now each of these objections is an objection which was raised back in the days of lore, when Richard Feynman was among the first to propose a quantum computer, and when most physicists and computer scientists didn't even know enough about quantum computing to shun it without learning it. But these objections, when you dig down and analyze them aren't much different from those of a different type of machine, a classical probabilistic machine. | Now each of these objections is an objection which was raised back in the days of lore, when Richard Feynman was among the first to propose a quantum computer, and when most physicists and computer scientists didn't even know enough about quantum computing to shun it without learning it. But these objections, when you dig down and analyze them aren't much different from those of a different type of machine, a classical probabilistic machine. | ||
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[[Image:Analogdigital.jpg|right|thumb|380px|A system can be (analog or digital) and (deterministic and probabilistic)]] | [[Image:Analogdigital.jpg|right|thumb|380px|A system can be (analog or digital) and (deterministic and probabilistic)]] | ||
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| + | Let me explain. First of all it is useful to define a few things. In particular it is useful to define digital and analog, or at least to define what I mean by these words. To me a digital system is a system whose configurations are a finite set. In other words a digital system has configurations which you might label as zero, one, two, etc. Now of course, most digital systems are abstractions. For example the digital information represented by voltages in your computer is digital in the sense that you define a certain voltage range to be a zero and another range to be one. Thus even though the underlying system may not be digital, from the perspective of how you use the computer, being digital is a good approximation. Analog systems, as opposed to digital systems, are the ones whose configurations aren't from a finite set. Thus, for example, in classical physics, consider the location of a bug on a line segment. To properly describe where the bug is we need to use the not finite set of real numbers. | ||
Okay, so digital and analog are words which we use to describe the configurations of a physical system (physicists would call these degrees of freedom.) But often a physical system, while it may be digital or analog, also has uncertainty associated with it. Thus it is useful to also make a distinction between systems which are deterministic and those which are probabilistic. Now technically these two ideas really refer to how a systems configurations change with time. But they also influence how we describe the system at a given time. A deterministic system is one in which the configuration at some time is completely determined by the configuration at some previous time. A probabilistic system is one in which the configurations change and, either because we are ignorant of information or for some more fundamental reason, these changes are not known with certainty. For systems which evolve deterministically, we can write down that we know the state of the system. For systems which evolve probabilistically, we aren't allowed to describe our system in such precise terms, but must associate probabilities to be in particular configurations. | Okay, so digital and analog are words which we use to describe the configurations of a physical system (physicists would call these degrees of freedom.) But often a physical system, while it may be digital or analog, also has uncertainty associated with it. Thus it is useful to also make a distinction between systems which are deterministic and those which are probabilistic. Now technically these two ideas really refer to how a systems configurations change with time. But they also influence how we describe the system at a given time. A deterministic system is one in which the configuration at some time is completely determined by the configuration at some previous time. A probabilistic system is one in which the configurations change and, either because we are ignorant of information or for some more fundamental reason, these changes are not known with certainty. For systems which evolve deterministically, we can write down that we know the state of the system. For systems which evolve probabilistically, we aren't allowed to describe our system in such precise terms, but must associate probabilities to be in particular configurations. | ||
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| + | [[Image:Allsortsofbits.jpg|right|thumb|380px|Deterministic bits, Probabilistic bits, and Quantum bits, a cheat sheet.]] | ||
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Now why am I being so pedantic in describing these words? Well because first of all there is often a confusion between analog systems and probabilistic systems. The first refers to degrees of freedom or configurations. The second refers to how a system changes in time, or what we can predict about it once we make a measurement on it. This category error has caused great confusion. Note that a system can be deterministic and digital, deterministic and analog, probabilistic and digital, or probabilistic and analog, but not, say, digital and analog. | Now why am I being so pedantic in describing these words? Well because first of all there is often a confusion between analog systems and probabilistic systems. The first refers to degrees of freedom or configurations. The second refers to how a system changes in time, or what we can predict about it once we make a measurement on it. This category error has caused great confusion. Note that a system can be deterministic and digital, deterministic and analog, probabilistic and digital, or probabilistic and analog, but not, say, digital and analog. | ||
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So what does this have to do with quantum computers? Well it turns out that the set made of deterministic and probabilistic really has another member, and this member is quantum. Quantum theory is the language which describes how systems evolve in time in our universe. Thus we are led, in our most fundamental physical theories, to descriptions of systems which are quantum mechanical. These systems can be analog or digital, but they are described in a quantum fashion. In particular, for probabilistic systems we described our configuration by a set of positive real numbers which sum to unity, while for quantum systems these numbers are replaced by amplitudes, complex numbers whose absolute value squared sum to unity. A quantum system is as simple as that: a system whose description is given by amplitudes and not probabilities. | So what does this have to do with quantum computers? Well it turns out that the set made of deterministic and probabilistic really has another member, and this member is quantum. Quantum theory is the language which describes how systems evolve in time in our universe. Thus we are led, in our most fundamental physical theories, to descriptions of systems which are quantum mechanical. These systems can be analog or digital, but they are described in a quantum fashion. In particular, for probabilistic systems we described our configuration by a set of positive real numbers which sum to unity, while for quantum systems these numbers are replaced by amplitudes, complex numbers whose absolute value squared sum to unity. A quantum system is as simple as that: a system whose description is given by amplitudes and not probabilities. | ||
Revision as of 04:18, 18 April 2008
Our confidence that the quest to build a large scale robust quantum computer is not a wild goose chase arises from the theory of fault-tolerant quantum computation. This theory tells us that if noise on a quantum system is weak enough, and our control over the quantum system is precise enough, then robust large scale quantum computation is possible with only a small (polylogarithmic) blow up in the size of the quantum circuits needed to achieve this robustness. In spite of this theory, however, there are considerable difficulties to the construction of physical systems which enact fault-tolerant quantum computation. In particular the device complexity of most fault-tolerant quantum error correcting routines is quite severe. Is there a better way to build a robust quantum computer than by using the complex and costly currently known fault-tolerant constructions?
We are persuing one possible alternative towards the standard method for constructing fault-tolerant quantum computers. The basic idea is motivated by examining why classical computers are able to perform fault-tolerant quantum computation. In particular we are investigating physical systems which have custom designed Hamiltonians such that the energy eigenvectors of these Hamiltonians are quantum error correcting codes and such that there is a free energy barrier preventing errors from ocuring on this information. Such systems would "self-correct" in much the same way that spin domains on classical hard drives self-correct errors when kept below a critical temperature. For a detailed explanation of what this project is about, scroll down below the references.
References
- D. Bacon, Operator Quantum Error-correcting Subsystems for Self-correcting Quantum Memories. Physical Review A, 73, 012340 (2006)
- D. Bacon, K.R. Brown, and K.B. Whaley, Coherence-Preserving Quantum Bits. Physical Review Letters, 87, 247902-1-247902-4 (2001)
- D. Bacon, Decoherence, Control, and Symmetry in Quantum Computers. Ph.D. thesis, University of California, Berkeley, CA (2001)
Contents
Lessons Unlearned
(creation of webpage in progress)
Quantum computing, at first sight, sounds like a hairbrained idea with absolutely no possible possibility of actually working in the real world. The reasons for this are plentiful, at least when you first start learning about quantum computers. Quantum states are described by a continuum of values. Uh, oh, that sounds like analog computing, which, if you really allow infinite precision, you will get unlimited power. Quantum gates are described by unitary matrices. Again these have a continuum of values. Quantum states "collapse" when you look at them. How do you quantum compute if simply looking at your computer destroys its computation?
Now each of these objections is an objection which was raised back in the days of lore, when Richard Feynman was among the first to propose a quantum computer, and when most physicists and computer scientists didn't even know enough about quantum computing to shun it without learning it. But these objections, when you dig down and analyze them aren't much different from those of a different type of machine, a classical probabilistic machine.
Let me explain. First of all it is useful to define a few things. In particular it is useful to define digital and analog, or at least to define what I mean by these words. To me a digital system is a system whose configurations are a finite set. In other words a digital system has configurations which you might label as zero, one, two, etc. Now of course, most digital systems are abstractions. For example the digital information represented by voltages in your computer is digital in the sense that you define a certain voltage range to be a zero and another range to be one. Thus even though the underlying system may not be digital, from the perspective of how you use the computer, being digital is a good approximation. Analog systems, as opposed to digital systems, are the ones whose configurations aren't from a finite set. Thus, for example, in classical physics, consider the location of a bug on a line segment. To properly describe where the bug is we need to use the not finite set of real numbers.
Okay, so digital and analog are words which we use to describe the configurations of a physical system (physicists would call these degrees of freedom.) But often a physical system, while it may be digital or analog, also has uncertainty associated with it. Thus it is useful to also make a distinction between systems which are deterministic and those which are probabilistic. Now technically these two ideas really refer to how a systems configurations change with time. But they also influence how we describe the system at a given time. A deterministic system is one in which the configuration at some time is completely determined by the configuration at some previous time. A probabilistic system is one in which the configurations change and, either because we are ignorant of information or for some more fundamental reason, these changes are not known with certainty. For systems which evolve deterministically, we can write down that we know the state of the system. For systems which evolve probabilistically, we aren't allowed to describe our system in such precise terms, but must associate probabilities to be in particular configurations.
Now why am I being so pedantic in describing these words? Well because first of all there is often a confusion between analog systems and probabilistic systems. The first refers to degrees of freedom or configurations. The second refers to how a system changes in time, or what we can predict about it once we make a measurement on it. This category error has caused great confusion. Note that a system can be deterministic and digital, deterministic and analog, probabilistic and digital, or probabilistic and analog, but not, say, digital and analog.
So what does this have to do with quantum computers? Well it turns out that the set made of deterministic and probabilistic really has another member, and this member is quantum. Quantum theory is the language which describes how systems evolve in time in our universe. Thus we are led, in our most fundamental physical theories, to descriptions of systems which are quantum mechanical. These systems can be analog or digital, but they are described in a quantum fashion. In particular, for probabilistic systems we described our configuration by a set of positive real numbers which sum to unity, while for quantum systems these numbers are replaced by amplitudes, complex numbers whose absolute value squared sum to unity. A quantum system is as simple as that: a system whose description is given by amplitudes and not probabilities.
Why Classical Computation is Possible?
Classical computers are both digital and deterministic. But if you go and take a microscope to your classical system what you will see isn't anything like these two categories. For example, if you go and look at the electrons in your silicon based transistor, not all of the electrons are doing the same things. Even in todays ultra pure system, real systems have defects (are dirty) and the electrons in the system are behaving in all sorts of strange ways. Of course the aggregate effect of all of these electrons bouncing around and doing all sorts of crazy things is, when digitized, deterministic. Thus the transistors in your computer are digital and deterministic in spite of the fact that the systems out of which they are constructed are both analog and probabilistic (or worse analog and quantum!)
