arXiv:quant-ph/0408119v1 19 Aug 2004
Quantum Computing and Hidden Variables II: The Complexity of
This paper shows that, if we could examine the entire history of a hidden variable, then we
could efficiently solve problems that are believed to be intractable even for quantum computers.
In particular, under any hidden-variable theory satisfying a reasonable axiom called “indiffer-
ence to the identity,” we could solve the Graph Isomorphism and Approximate Shortest Vector
problems in polynomial time, as well as an oracle problem that is known to require quantum
exponential time. We could also search an N-item database using O?N1/3?queries, as opposed
queries with Grover’s search algorithm.
optimal, meaning that we could probably not solve NP-complete problems in polynomial time.
We thus obtain the first good example of a model of computation that appears slightly more
powerful than the quantum computing model.
On the other hand, the N1/3bound is
It is often stressed that hidden-variable theories, such as Bohmian mechanics, yield exactly the
same predictions as ordinary quantum mechanics.
different picture of physical reality, with an additional layer of dynamics beyond that of a state
vector evolving unitarily. This paper addresses a question that, to our knowledge, had never been
raised before: what is the computational complexity of simulating that additional dynamics?
other words, if we could examine a hidden variable’s entire history, then could we solve problems
in polynomial time that are intractable even for quantum computers?
We present strong evidence that the answer is yes.
whether two graphs G and H are isomorphic; while given a basis for a lattice L ∈ Rn, the Approx-
imate Shortest Vector problem asks for a nonzero vector in L within a√n factor of the shortest
one. We show that both problems are efficiently solvable by sampling a hidden variable’s history,
provided the hidden-variable theory satisfies a reasonable axiom that we call “indifference to the
identity operation.” By contrast, despite a decade of effort, neither problem is known to lie in
BQP, the class of problems solvable in quantum polynomial time with bounded error probability.1
Thus, if we let DQP (Dynamical Quantum Polynomial-Time) be the class of problems solvable in
our new model, then this already provides circumstantial evidence that BQP is strictly contained
However, the evidence is stronger than this. For we actually show that DQP contains an entire
class of problems, of which Graph Isomorphism and Approximate Shortest Vector are special
On the other hand, these theories describe a
The Graph Isomorphism problem asks
∗University of California, Berkeley. Email: firstname.lastname@example.org.
1See www.complexityzoo.com for more information about the complexity classes mentioned in this paper.
cases. Computer scientists know this class as Statistical Zero Knowledge, or SZK.
in previous work  we showed that “relative to an oracle,” SZK is not contained in BQP. This is
a technical concept implying that any proof of SZK ⊆ BQP would require techniques unlike those
that are currently known. Combining our result that SZK ⊆ DQP with the oracle separation of ,
we obtain that BQP ?= DQP relative to an oracle as well. Given computer scientists’ longstanding
inability to separate basic complexity classes, this is nearly the best evidence one could hope for
that sampling histories yields more power than standard quantum computation.
Besides solving SZK problems, we also show that by sampling histories, one could search an
unordered database of N items for a single “marked item” using only O?N1/3?database queries.
algorithms require Θ(N) queries.2On the other hand, we also show that our N1/3upper bound is
the best possible—so even in the histories model, one cannot search an N-item database in (logN)c
steps for some fixed power c. This implies that NP ?⊂ DQP relative to an oracle, which in turn
suggests that DQP is still not powerful enough to solve NP-complete problems in polynomial time.
Note that while Graph Isomorphism and Approximate Shortest Vector are in NP, it is strongly
believed that they are not NP-complete.
At this point we should address a concern that many readers will have.
quantum mechanics by positing the “unphysical” ability to sample histories, isn’t it completely
unsurprising if we can then solve problems that were previously intractable?
answer is no, for three reasons.
First, almost every change that makes the quantum computing model more powerful, seems
to make it so much more powerful that NP-complete and even harder problems become solvable
efficiently. To give some examples, NP-complete problems can be solved in polynomial time using
a nonlinear Schr¨ odinger equation, as shown by Abrams and Lloyd ; using closed timelike curves,
as shown by Bacon ; or using a measurement rule of the form |ψ|pfor any p ?= 2, as shown by
us . It is also easy to see that we could solve NP-complete problems if, given a quantum state
|ψ?, we could request a classical description of |ψ?, such as a list of amplitudes or a preparation
By contrast, ours is the first independently motivated model we know of that seems
more powerful than quantum computing, but only slightly so.4
unordered search in our model takes about N1/3steps, as compared to N steps classically and
N1/2quantum-mechanically, suggests that DQP somehow “continues a sequence” that begins with
P and BQP. It would be interesting to find a model in which search takes N1/4or N1/5steps.
The second reason our results are surprising is that, given a hidden variable, the distribution
over its possible values at any single time is governed by standard quantum mechanics, and is
therefore efficiently samplable on a quantum computer.
confers any extra computational power, then it can only be because of correlations between the
variable’s values at different times.
The third reason is our criterion for success.
Graph Isomorphism under some hidden-variable theory; or even that, under any theory satisfying
By comparison, Grover’s quantum search algorithm  requires Θ?N1/2?queries, while classical
Once we extend
We believe the
Moreover, the striking fact that
So if examining the variable’s history
We are not saying merely that one can solve
2For readers unfamiliar with asymptotic notation: O(f (N)) means “at most order f (N),” Ω(f (N)) means “at
least order f (N),” and Θ(f (N)) means “exactly order f (N).”
3For as Abrams and Lloyd  observed, we can so arrange things that |ψ? = |0? if an NP-complete instance of
interest to us has no solution, but |ψ? =√1 − ε|0? +√ε|1? for some tiny ε if it has a solution.
4One can define other, less motivated, models with the same property by allowing “non-collapsing measurements”
of quantum states, but these models are very closely related to ours. Indeed, a key ingredient of our results will be
to show that certain kinds of non-collapsing measurements can be simulated using histories.
The trouble is that Bohmian mechanics does not quite fit in our framework: as discussed in ,
we cannot have deterministic hidden-variable trajectories for discrete degrees of freedom such as
qubits. Even worse, Bohmian mechanics violates the continuous analogue of the indifference
axiom.On the other hand, this means that by trying to implement (say) the juggle subroutine
with Bohmian trajectories, one might learn not only about Bohmian mechanics and its relation
to quantum computation, but also about how essential the indifference axiom really is for our
On the computer science side, a key open problem is to show better upper bounds on DQP.
Recall that we were only able to show DQP ⊆ EXP, by giving a classical exponential-time algorithm
to simulate the flow theory FT . Can we improve this to (say) DQP ⊆ PSPACE? Clearly it would
suffice to give a PSPACE algorithm that computes the transition probabilities for some theory T
satisfying the indifference and robustness axioms. On the other hand, this might not be necessary—
that is, there might be an indirect simulation method that does not work by computing (or even
sampling from) the distribution over histories. It would also be nice to pin down the complexities
of simulating specific hidden-variable theories, such as FT and ST .
I thank Umesh Vazirani, Ronald de Wolf, and an anonymous reviewer for comments on an earlier
version of this paper; Antony Valentini and Rob Spekkens for helpful discussions; and Andris
Ambainis for correcting an ambiguity in the definition of DQP. Supported by an NSF Graduate
Fellowship and by DARPA grant F30602-01-2-0524.
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